Lectures on Modular Forms and Hecke Operators

Kenneth A. Ribet  William A. Stein

December 2, 2011

ii

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 The Main Objects  3
1.1 Torsion points on elliptic curves . . . . . . . . . . . . . . . . . . . . 3
1.1.1 The Tate module . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Modular Representations and Algebraic Curves  7
2.1 Modular forms and Arithmetic . . . . . . . . . . . . . . . . . . . . 7
2.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Parity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Conjectures of Serre (mod ℓ version) . . . . . . . . . . . . . . . . . 9
2.5 General remarks on mod p Galois representations . . . . . . . . . . 10
2.6 Serre’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Wiles’s perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Modular Forms of Level 1  13
3.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Some examples and conjectures . . . . . . . . . . . . . . . . . . . . 14
3.3 Modular forms as functions on lattices . . . . . . . . . . . . . . . . 15
3.4 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4.1 Relations Between Hecke Operators . . . . . . . . . . . . . 18
3.5 Hecke operators directly on q-expansions . . . . . . . . . . . . . . . 19
3.5.1 Explicit description of sublattices . . . . . . . . . . . . . . . 20
3.5.2 Hecke operators on q-expansions . . . . . . . . . . . . . . . 21
3.5.3 The Hecke algebra and eigenforms . . . . . . . . . . . . . . 22
3.5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Two Conjectures about Hecke operators on level 1 modular forms . 24
3.6.1 Maeda’s conjecture . . . . . . . . . . . . . . . . . . . . . . . 24
3.6.2 The Gouvea-Mazur conjecture . . . . . . . . . . . . . . . . 24
3.7 An Algorithm for computing characteristic polynomials of Hecke operators 25
3.7.1 Review of basic facts about modular forms . . . . . . . . . 26
3.7.2 The Naive approach . . . . . . . . . . . . . . . . . . . . . . 26
3.7.3 The Eigenform method . . . . . . . . . . . . . . . . . . . . 27
3.7.4 How to write down an eigenvector over an extension field . 28
3.7.5 Simple example: weight 36, p = 3 . . . . . . . . . . . . . . . 29
4 Duality, Rationality, and Integrality  31
4.1 Modular forms for SL_{2}(Z) and Eisenstein series . . . . . . . . . . . 31
4.2 Pairings between Hecke algebras and modular forms . . . . . . . . 32
4.3 Eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5 A Result from Victor Miller’s thesis . . . . . . . . . . . . . . . . . 34
4.6 The Petersson inner product . . . . . . . . . . . . . . . . . . . . . 35
5 Analytic Theory of Modular Curves  39
5.1 The Modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.1 The Upper half plane . . . . . . . . . . . . . . . . . . . . . 39
5.2 Points on modular curves parameterize elliptic curves with extra structure 40
5.3 The Genus of X(N) . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 Modular Curves  47
6.1 Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Modular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3 Classifying Γ(N)-structures . . . . . . . . . . . . . . . . . . . . . . 48
6.4 More on integral Hecke operators . . . . . . . . . . . . . . . . . . . 49
6.5 Complex conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.6 Isomorphism in the real case . . . . . . . . . . . . . . . . . . . . . 49
6.7 The Eichler-Shimura isomorphism . . . . . . . . . . . . . . . . . . 50
7 Modular Symbols  53
7.1 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.2.1 Using continued fractions to obtain surjectivity . . . . . . . 55
7.2.2 Triangulating X(G) to obtain injectivity . . . . . . . . . . . 56
7.3 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.4 Modular symbols and rational homology . . . . . . . . . . . . . . . 60
7.5 Special values of L-functions . . . . . . . . . . . . . . . . . . . . . . 61
8 Modular Forms of Higher Level  65
8.1 Modular Forms on Γ_{1}(N) . . . . . . . . . . . . . . . . . . . . . . . 65
8.2 Diamond bracket and Hecke operators . . . . . . . . . . . . . . . . 66
8.2.1 Diamond bracket operators . . . . . . . . . . . . . . . . . . 66
8.2.2 Hecke operators on q-expansions . . . . . . . . . . . . . . . 68
8.3 Old and new subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 68
9 Newforms and Euler Products  71

9.1 Atkin-Lehner-Li theory . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2 The U_{p} operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.2.1 A Connection with Galois representations . . . . . . . . . . 76
9.2.2 When is U_{p} semisimple? . . . . . . . . . . . . . . . . . . . . 77
9.2.3 An Example of non-semisimple U_{p} . . . . . . . . . . . . . . 77
9.3 The Cusp forms are free of rank 1 over T_{C} . . . . . . . . . . . . . 77
9.3.1 Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.3.2 General level . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.4 Decomposing the anemic Hecke algebra . . . . . . . . . . . . . . . 80
10 Some Explicit Genus Computations  83
10.1 Computing the dimension of S_{2}(Γ) . . . . . . . . . . . . . . . . . . 83
10.2 Application of Riemann-Hurwitz . . . . . . . . . . . . . . . . . . . 84
10.3 The Genus of X(N) . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.4 The Genus of X_{0}(N), for N prime . . . . . . . . . . . . . . . . . . 86
10.5 Modular forms mod p . . . . . . . . . . . . . . . . . . . . . . . . . 86
11 The Field of Moduli  89
11.1 Algebraic definition of X(N) . . . . . . . . . . . . . . . . . . . . . 89
11.2 Digression on moduli . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11.3 When is ρ_{E} surjective? . . . . . . . . . . . . . . . . . . . . . . . . . 91
11.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
11.5 A descent problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
11.6 Second look at the descent exercise . . . . . . . . . . . . . . . . . . 94
11.7 Action of GL_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
12 Hecke Operators as Correspondences  97
12.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
12.2 Maps induced by correspondences . . . . . . . . . . . . . . . . . . . 99
12.3 Induced maps on Jacobians of curves . . . . . . . . . . . . . . . . . 100
12.4 More on Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . 100
12.5 Hecke operators acting on Jacobians . . . . . . . . . . . . . . . . . 101
12.5.1 The Albanese Map . . . . . . . . . . . . . . . . . . . . . . . 102
12.5.2 The Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . 103
12.6 The Eichler-Shimura relation . . . . . . . . . . . . . . . . . . . . . 103
12.7 Applications of the Eichler-Shimura relation . . . . . . . . . . . . . 107
12.7.1 The Characteristic polynomial of Frobenius . . . . . . . . . 107
12.7.2 The Cardinality of J_{0}(N)(F_{p}) . . . . . . . . . . . . . . . . . 109
13 Abelian Varieties  111
13.1 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
13.2 Complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
13.2.1 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 112
13.2.2 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
13.2.3 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 115
13.3 Abelian varieties as complex tori . . . . . . . . . . . . . . . . . . . 115
13.3.1 Hermitian and Riemann forms . . . . . . . . . . . . . . . . 116
13.3.2 Complements, quotients, and semisimplicity of the endomorphism algebra117
13.3.3 Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . 119
13.4 A Summary of duality and polarizations . . . . . . . . . . . . . . . 119

13.4.1 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
13.4.2 The Picard group . . . . . . . . . . . . . . . . . . . . . . . . 120
13.4.3 The Dual as a complex torus . . . . . . . . . . . . . . . . . 120
13.4.4 The Néron-Severi group and polarizations . . . . . . . . . . 121
13.4.5 The Dual is functorial . . . . . . . . . . . . . . . . . . . . . 121
13.5 Jacobians of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
13.5.1 Divisors on curves and linear equivalence . . . . . . . . . . 122
13.5.2 Algebraic definition of the Jacobian . . . . . . . . . . . . . 123
13.5.3 The Abel-Jacobi theorem . . . . . . . . . . . . . . . . . . . 124
13.5.4 Every abelian variety is a quotient of a Jacobian . . . . . . 125
13.6 Néron models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
13.6.1 What are Néron models? . . . . . . . . . . . . . . . . . . . 127
13.6.2 The Birch and Swinnerton-Dyer conjecture and Néron models129
13.6.3 Functorial properties of Neron models . . . . . . . . . . . . 131
14 Abelian Varieties Attached to Modular Forms  133
14.1 Decomposition of the Hecke algebra . . . . . . . . . . . . . . . . . 133
14.1.1 The Dimension of the algebras L_{f} . . . . . . . . . . . . . . 134
14.2 Decomposition of J_{1}(N) . . . . . . . . . . . . . . . . . . . . . . . . 135
14.2.1 Aside: intersections and congruences . . . . . . . . . . . . . 136
14.3 Galois representations attached to A_{f} . . . . . . . . . . . . . . . . 137
14.3.1 The Weil pairing . . . . . . . . . . . . . . . . . . . . . . . . 138
14.3.2 The Determinant . . . . . . . . . . . . . . . . . . . . . . . . 140
14.4 Remarks about the modular polarization . . . . . . . . . . . . . . . 141
15 Modularity of Abelian Varieties  143
15.1 Modularity over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.2 Modularity of elliptic curves over \overline{Q} . . . . . . . . . . . . . . . . . 146
15.3 Modularity of abelian varieties over \overline{Q} . . . . . . . . . . . . . . . . 146
16 L-functions  149
16.1 L-functions attached to modular forms . . . . . . . . . . . . . . . . 149
16.1.1 Analytic continuation and functional equations . . . . . . . 150
16.1.2 A Conjecture about nonvanishing of L(f, k/2) . . . . . . . . 152
16.1.3 Euler products . . . . . . . . . . . . . . . . . . . . . . . . . 152
16.1.4 Visualizing L-function . . . . . . . . . . . . . . . . . . . . . 153
17 The Birch and Swinnerton-Dyer Conjecture  155
17.1 The Rank conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 155
17.2 Refined rank zero conjecture . . . . . . . . . . . . . . . . . . . . . . 157
17.2.1 The Number of real components . . . . . . . . . . . . . . . 158
17.2.2 The Manin index . . . . . . . . . . . . . . . . . . . . . . . . 158
17.2.3 The Real volume Ω_{A} . . . . . . . . . . . . . . . . . . . . . . 159
17.2.4 The Period mapping . . . . . . . . . . . . . . . . . . . . . . 160
17.2.5 The Manin-Drinfeld theorem . . . . . . . . . . . . . . . . . 160
17.2.6 The Period lattice . . . . . . . . . . . . . . . . . . . . . . . 160
17.2.7 The Special value L(A, 1) . . . . . . . . . . . . . . . . . . . 161
17.2.8 Rationality of L(A, 1)/Ω_{A} . . . . . . . . . . . . . . . . . . . 161
17.3 General refined conjecture . . . . . . . . . . . . . . . . . . . . . . . 163
17.4 The Conjecture for non-modular abelian varieties . . . . . . . . . . 163

17.5 Visibility of Shafarevich-Tate groups . . . . . . . . . . . . . . . . . 164
17.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
17.5.2 Every element of H^{1}(K, A) is visible somewhere . . . . . . . 166
17.5.3 Visibility in the context of modularity . . . . . . . . . . . . 166
17.5.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . 168
17.6 Kolyvagin’s Euler system of Heegner points . . . . . . . . . . . . . 169
17.6.1 A Heegner point when N = 11 . . . . . . . . . . . . . . . . 178
17.6.2 Kolyvagin’s Euler system for curves of rank at least 2 … 179
18 The Gorenstein Property for Hecke Algebras  181
18.1 Mod ℓ representations associated to modular forms . . . . . . . . . 181
18.2 The Gorenstein property . . . . . . . . . . . . . . . . . . . . . . . . 184
18.3 Proof of the Gorenstein property . . . . . . . . . . . . . . . . . . . 186
18.3.1 Vague comments . . . . . . . . . . . . . . . . . . . . . . . . 189
18.4 Finite flat group schemes . . . . . . . . . . . . . . . . . . . . . . . 190
18.5 Reformulation of V = W problem . . . . . . . . . . . . . . . . . . . 190
18.6 Dieudonné theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
18.7 The proof: part II . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
18.8 Key result of Boston-Lenstra-Ribet . . . . . . . . . . . . . . . . . . 194
19 Local Properties of ρ_{λ}  197
19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
19.2 Local properties at primes p ∤ N . . . . . . . . . . . . . . . . . . . 198
19.3 Weil-Deligne Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 198
19.4 Local properties at primes p | N . . . . . . . . . . . . . . . . . . . 198
19.5 Definition of the reduced conductor . . . . . . . . . . . . . . . . . . 199
20 Adelic Representations  201
20.1 Adelic representations associated to modular forms . . . . . . . . . 201
20.2 More local properties of the ρ_{λ}. . . . . . . . . . . . . . . . . . . . . 204
20.2.1 Possibilities for π_{p} . . . . . . . . . . . . . . . . . . . . . . . 205
20.2.2 The case ℓ = p . . . . . . . . . . . . . . . . . . . . . . . . . 206
20.2.3 Tate curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
21 Serre’s Conjecture  209
21.1 The Family of λ-adic representations attached to a newform …. 210
21.2 Serre’s Conjecture A . . . . . . . . . . . . . . . . . . . . . . . . . . 210
21.2.1 The Field of definition of ρ . . . . . . . . . . . . . . . . . . 211
21.3 Serre’s Conjecture B . . . . . . . . . . . . . . . . . . . . . . . . . . 212
21.4 The Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
21.4.1 Remark on the case N(ρ) = 1 . . . . . . . . . . . . . . . . . 213
21.4.2 Remark on the proof of Conjecture B . . . . . . . . . . . . 214
21.5 The Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
21.5.1 The Weight modulo ℓ − 1 . . . . . . . . . . . . . . . . . . . 215
21.5.2 Tameness at ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . 215
21.5.3 Fundamental characters of the tame extension . . . . . . . 216
21.5.4 The Pair of characters associated to ρ . . . . . . . . . . . . 217
21.5.5 Recipe for the weight . . . . . . . . . . . . . . . . . . . . . 218
21.5.6 The World’s first view of fundamental characters . . . . . . 219
21.5.7 Fontaine’s theorem . . . . . . . . . . . . . . . . . . . . . . . 219

21.5.8 Guessing the weight (level 2 case) . . . . . . . . . . . . . . 219
21.5.9 θ-cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
21.5.10 Edixhoven’s paper . . . . . . . . . . . . . . . . . . . . . . . 222
21.6 The Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
21.6.1 A Counterexample . . . . . . . . . . . . . . . . . . . . . . . 224
21.7 The Weight revisited: level 1 case . . . . . . . . . . . . . . . . . . . 225
21.7.1 Companion forms . . . . . . . . . . . . . . . . . . . . . . . . 225
21.7.2 The Weight: the remaining level 1 case . . . . . . . . . . . . 226
21.7.3 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
22 Fermat’s Last Theorem  229
22.1 The application to Fermat . . . . . . . . . . . . . . . . . . . . . . . 229
22.2 Modular elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 231
23 Deformations  233
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
23.2 Condition (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
23.2.1 Finite flat representations . . . . . . . . . . . . . . . . . . . 235
23.3 Classes of liftings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
23.3.1 The case p ≠ ℓ . . . . . . . . . . . . . . . . . . . . . . . . . 235
23.3.2 The case p = ℓ . . . . . . . . . . . . . . . . . . . . . . . . . 236
23.4 Wiles’s Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . 237
24 The Hecke Algebra T_{Σ}  239
24.1 The Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
24.2 The Maximal ideal in R . . . . . . . . . . . . . . . . . . . . . . . . 241
24.2.1 Strip away certain Euler factors . . . . . . . . . . . . . . . . 241
24.2.2 Make into an eigenform for U_{ℓ} . . . . . . . . . . . . . . . . 242
24.3 The Galois representation . . . . . . . . . . . . . . . . . . . . . . . 243
24.3.1 The Structure of T_{𝔪} . . . . . . . . . . . . . . . . . . . . . . 244
24.3.2 The Philosophy in this picture . . . . . . . . . . . . . . . . 244
24.3.3 Massage ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
24.3.4 Massage ρ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
24.3.5 Representations from modular forms mod ℓ . . . . . . . . . 246
24.3.6 Representations from modular forms mod ℓ^{n} . . . . . . . . 246
24.4 ρ′ is of type Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
24.5 Isomorphism between T_{𝔪} and R_{𝔪_{R}} . . . . . . . . . . . . . . . . . . 248
24.6 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
24.7 Wiles’s main conjecture . . . . . . . . . . . . . . . . . . . . . . . . 250
24.8 T_{Σ} is a complete intersection . . . . . . . . . . . . . . . . . . . . . 252
24.9 The Inequality #O/η ≤ #℘_{T} /℘_{T}^{2} ≤ #℘_{R}/℘_{R}^{2} . . . . . . . . . . . . 252
24.9.1 The Definitions of the ideals . . . . . . . . . . . . . . . . . . 253
24.9.2 Aside: Selmer groups . . . . . . . . . . . . . . . . . . . . . . 254
24.9.3 Outline of some proofs . . . . . . . . . . . . . . . . . . . . . 254
25 Computing with Modular Forms and Abelian Varieties  257
26 The Modular Curve X_{0}(389)  259
26.1 Factors of J_{0}(389) . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
26.1.1 Newforms of level 389 . . . . . . . . . . . . . . . . . . . . . 260

26.1.2 Isogeny structure . . . . . . . . . . . . . . . . . . . . . . . . 261
26.1.3 Mordell-Weil ranks . . . . . . . . . . . . . . . . . . . . . . . 261
26.2 The Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
26.2.1 The Discriminant is divisible by p . . . . . . . . . . . . . . 262
26.2.2 Congruences primes in S_{p+1}(Γ_{0}(1)) . . . . . . . . . . . . . . 263
26.3 Supersingular points in characteristic 389 . . . . . . . . . . . . . . 264
26.3.1 The Supersingular j-invariants in characteristic 389 …. 264
26.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
26.4.1 The Shafarevich-Tate group . . . . . . . . . . . . . . . . . . 264
26.4.2 Weierstrass points on X_{0}^{+}(p) . . . . . . . . . . . . . . . . . 265
26.4.3 A Property of the plus part of the integral homology …. 265
26.4.4 The Field generated by points of small prime order on an elliptic curve265
References  267

Preface

This book began when the second author typed notes for the first author’s 1996 Berkeley course on modular forms with a view toward explaining some of the key ideas in Wiles’s celebrated proof of Fermat’s Last Theorem. The second author then expanded and rewrote the notes while teaching a course at Harvard in 2003 on modular abelian varieties.

The intended audience of this book is advanced graduate students and mathematical researchers. This book is more advanced than [LR11, Ste07b, DS05], and at a relatively similar level to [DI95], though with more details. It should be substantially more accessible than a typical research paper in the area.

Some things about how this book is (or will be!) fully “modern” in that it takes into account:

• The full modularity theorem.

• The proof of the full Serre conjecture

• Computational techniques (algorithms, Sage)

Notation

≅ an isomorphism ≈ a noncanonical isomorphism

Acknowledgement

Joe Wetherell wrote the first version of Section 2.5.

Contact

Kenneth A. Ribet (ribet@math.berkeley.edu) William A. Stein (wstein@gmail.com)

The Main Objects

1.1 Torsion points on elliptic curves

The main geometric objects that we will study in this book are elliptic curves, which are curves of genus one equipped with a distinguished point. More generally, we consider certain algebraic curves of larger genus called modular curves, which in turn give rise via the Jacobian construction to higher-dimensional abelian varieties from which we will obtain representations of the Galois group Gal(\overline{Q}/Q) of the rational numbers.

It is convenient to view the group of complex points E(C) on an elliptic curve E over the complex numbers C as a quotient C/L. Here

L = {∫_{γ} ω : γ ∈ H_{1}(E(C), Z)}

is a lattice attached to a nonzero holomorphic differential ω on E, and the homology H_{1}(E(C), Z) ≈ Z×Z is the abelian group of smooth closed paths on E(C) modulo the homology relations.

Viewing E as C/L immediately gives us information about the structure of the group of torsion points on E, which we exploit in the next section to construct two-dimensional representations of Gal(\overline{Q}/Q).

1.1.1 The Tate module

In the 1940s, Andre Weil studied the analogous situation for elliptic curves defined over a finite field k. He desperately wanted to find an algebraic way to describe the above relationship between elliptic curves and lattices. He found an algebraic definition of L/nL, when n is prime to the characteristic of k.

Let

E[n] := \{P ∈ E(\overline{k}) : nP = 0\}.

When E is defined over C,

E[n] = (\frac{1}{n}L) /L ≅ L/nL ≈ (Z/nZ) × (Z/nZ),

so E[n] is a purely algebraic object canonically isomorphic to L/nL.

Now suppose E is defined over an arbitrary field k. For any prime ℓ, let

E[ℓ^{∞}] := \{P ∈ E(\overline{k}) : ℓ^{ν}P = 0, some ν ≥ 1\}
= ⋃_{ν=1}^{∞} E[ℓ^{ν}] = \underrightarrow{lim} E[ℓ^{ν}].

In an analogous way, Tate constructed a rank 2 free Z_{ℓ}-module

T_{ℓ}(E) := \underleftarrow{lim} E[ℓ^{ν}],

where the map E[ℓ^{ν}] → E[ℓ^{ν−1}] is multiplication by ℓ. The Z/ℓ^{ν}Z-module structure of E[ℓ^{ν}] is compatible with the maps E[ℓ^{ν}] ⟶^{ℓ} E[ℓ^{ν−1}] (see, e.g., [Sil92, III.7]). If ℓ is coprime to the characteristic of the base field k, then T_{ℓ}(E) is free of rank 2 over Z_{ℓ}, and

V_{ℓ}(E) := T_{ℓ}(E) ⊗ Q_{ℓ}

is a 2-dimensional vector space over Q_{ℓ}.

1.2 Galois representations

Number theory is largely concerned with the Galois group Gal(\overline{Q}/Q), which is often studied by considering continuous linear representations

ρ : Gal(\overline{Q}/Q) → GL_{n}(K)

where K is a field and n is a positive integer, usually 2 in this book. Artin, Shimura, Taniyama, and Tate pioneered the study of such representations.

Let E be an elliptic curve defined over the rational numbers Q. Then Gal(\overline{Q}/Q) acts on the set E[n], and this action respects the group operations, so we obtain a representation

ρ : Gal(\overline{Q}/Q) → Aut(E[n]) ≈ GL_{2}(Z/nZ).

Let K be the field cut out by the ker(ρ), i.e., the fixed field of ker(ρ). Then K is a finite Galois extension of Q. Since

Gal(K/Q) ≅ Gal(\overline{Q}/Q)/ ker ρ ≅ Im ρ ↪ GL_{2}(Z/nZ)

we obtain, in this way, subgroups of GL_{2}(Z/nZ) as Galois groups.

Shimura showed that if we start with the elliptic curve E defined by the equation y^{2} + y = x^{3} − x^{2} then for “most” n the image of ρ is all of GL_{2}(Z/nZ). More generally, the image is “most” of GL_{2}(Z/nZ) when E does not have complex multiplication. (We say E has complex multiplication if its endomorphism ring over C is strictly larger than Z.)

1.3 Modular forms

Many spectacular theorems and deep conjectures link Galois representations with modular forms. Modular forms are extremely symmetric analytic objects, which we will first view as holomorphic functions on the complex upper half plane that behave well with respect to certain groups of transformations.

Let SL_{2}(Z) be the group of 2 × 2 integer matrices with determinant 1. For any positive integer N, consider the subgroup

Γ_{1}(N) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z) : a ≡ d ≡ 1, c ≡ 0 (mod N)\}

of matrices in SL_{2}(Z) that are of the form (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}) when reduced modulo N.

The space S_{k}(N) of cusp forms of weight k and level N for Γ_{1}(N) consists of all holomorphic functions f(z) on the complex upper half plane

𝔥 = {z ∈ C : Im(z) > 0}

that vanish at the cusps (see below) and satisfy the equation

f (\frac{az + b}{cz + d}) = (cz + d)^{k}f(z) for all (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{1}(N) and z ∈ 𝔥.

Thus f(z + 1) = f(z), so f determines a function F of q(z) = e^{2πiz} such that F (q) = f(z). Viewing F as a function on {z : 0 < |z| < 1}, the condition that f(z) is holomorphic and vanishes at infinity is that F (z) extends to a holomorphic function on {z : |z| < 1} and F (0) = 0. In this case, f is determined by its Fourier expansion

f(q) = ∑_{n=1}^{∞} a_{n}q^{n}.

It is also useful to consider the space M_{k}(N) of modular forms of level N, which is defined in the same way as S_{k}(N), except that the condition that F (0) = 0 is relaxed, and we require only that F extends to a holomorphic function at 0 (and there is a similar condition at the cusps other than ∞).

The spaces M_{k}(N) and S_{k}(N) are finite dimensional. Example 1.3.1. We compute dim(M_{5}(30)) and dim(S_{5}(30)) in Sage:

sage: ModularForms(Gamma1(30),5).dimension()

112

sage: CuspForms(Gamma1(30),5).dimension()

80

For example, the space S_{12}(1) has dimension one and is spanned by the famous cusp form

Δ = q ∏_{n=1}^{∞}(1 − q^{n})^{24} = ∑_{n=1}^{∞} τ(n)q^{n}.

The coefficients τ(n) define the Ramanujan τ-function. A non-obvious fact is that τ is multiplicative and for every prime p and positive integer ν, we have

τ(p^{ν+1}) = τ(p)τ(p^{ν}) − p^{11}τ(p^{ν−1}).

Example 1.3.2. We draw a plot of the Δ function (using 20 terms of the qexpansion) on the upper half plane. Notice the symmetry Δ(z) = Δ(z + 1):

sage: z = var('z'); q = exp(2*pi*i*z)
sage: D = delta_qexp(20)(q)
sage: complex_plot(D, (-2,2), (0,.75),  plot_points=200)

1.4 Hecke operators

Mordell defined operators T_{n}, n ≥ 1, on S_{k}(N) which are called Hecke operators. These proved very fruitful. The set of such operators forms a commuting family of endomorphisms and is hence “almost” simultaneously diagonalizable.

Often there is a basis f_{1}, …, f_{r} of S_{k}(N) such that each f = f_{i} = ∑_{n=1}^{∞} a_{n}q^{n} is a simultaneous eigenvector for all the Hecke operators T_{n} normalized so that T_{n}f = a_{n}f, i.e., so the coefficient of q is 1. In this situation, the eigenvalues a_{n} are necessarily algebraic integers and the field

K = K_{f} = Q(…, a_{n}, …)

generated by all a_{n} is finite over Q.

The a_{n} exhibit remarkable properties. For example,

τ(n) ≡ ∑_{d|n} d^{11} (mod 691).

We can check this congruence for n = 30 in Sage as follows:

sage: n=30; t = delta_qexp(n+1)[n]; t

-29211840

sage: sigma(n,11)

17723450167663752

sage: (sigma(n,11) - t)%691

0

The key to studying and interpreting the a_{n} is to understand the deep connections between Galois representations and modular forms that were discovered by Serre, Shimura, Eichler and Deligne.

Modular Representations and Algebraic Curves

2.1 Modular forms and Arithmetic

Consider a cusp form

f = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{k}(N)

which is an eigenform for all of the Hecke operators T_{p}, and assume f is normalized so a_{1} = 1. Then the the L-function

L(f, s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

Hecke proved that L(f, s) extends uniquely to a holomorphic function on C that satisfies a functional equation. He did this using the (Mellin) integral transform

∫_{0}^{∞} f(it)t^{s} \frac{dt}{t} = N^{s/2}(2π)^{−s}Γ(s)L(f, s) = Λ(f, s).

There is a complex number c with absolute value 1 such that the function Λ(f, s) satisfies the functional equation

Λ(f, s) = cΛ(f̃, k − s),

where f̃ = ∑_{n=1}^{∞} \overline{a}_{n}q^{n} ∈ S_{k}(N) is obtained from f by complex conjugation of coefficients (see Section 6.5).

Let K = Q(a_{1}, a_{2}, …) be the number field generated by the Fourier coefficients of f. One can show that the a_{n} are algebraic integers and that K is a number field. When k = 2, Shimura associated to f an abelian variety A_{f} over Q of dimension [K : Q] on which Z[a_{1}, a_{2}, …] acts [Shi94, Theorem 7.14].

Example 2.1.1 (Modular Elliptic Curves). Suppose now that all coefficients a_{n} of f lie in Q so that [K : Q] = 1 and hence A_{f} is a one dimensional abelian variety, i.e., an elliptic curve. An elliptic curve isogenous to one arising via this construction is called modular.

Definition 2.1.2. Elliptic curves E_{1} and E_{2} are isogenous if there is a morphism E_{1} → E_{2} of algebraic groups, having finite kernel.

The following theorem motivates much of the theory discussed in this course. It is a theorem of Breuil, Conrad, Diamond, Taylor, and Wiles (see [BCDT01]).

Theorem 2.1.3 (Modularity Theorem). Every elliptic curve over Q is modular, that is, isogenous to a curve constructed in the above way.

For k ≥ 2 Serre and Deligne discovered a way to associate to f a family of ℓ-adic representations. Let ℓ be a prime number and K = Q(a_{1}, a_{2}, …) be as above. Then it is well known that

K ⊗_{Q} Q_{ℓ} ≅ ∏_{λ|ℓ} K_{λ}.

One can associate to f a representation

ρ_{ℓ,f} : G = Gal(\overline{Q}/Q) → GL_{2}(K ⊗_{Q} Q_{ℓ})

unramified at all primes p ∤ ℓN. Let \overline{Z} be the ring of all algebraic integers. For ρ_{ℓ,f} to be unramified at p we mean that for all primes P of \overline{Z} lying over p, the inertia subgroup of the decomposition group at P is contained in the kernel of ρ_{ℓ,f}. The decomposition group D_{P} at P is the set of those g ∈ G which fix P. Let \overline{k} be the residue field \overline{Z}/P and k = F_{p}. Then the inertia group I_{P} is the kernel of the surjective map D_{P} → Gal(\overline{k}/k).

Now I_{P} ⊂ D_{P} ⊂ Gal(\overline{Q}/Q) and D_{P} /I_{P} is pro-cyclic (being isomorphic to the Galois group Gal(\overline{k}/k)), so it is generated by a Frobenious automorphism Frob_{p} lying over p. One has

tr(ρ_{ℓ,f}(Frob_{p})) = a_{p} ∈ K ⊂ K ⊗ Q_{ℓ}

and

det(ρ_{ℓ,f}) = χ_{ℓ}^{k−1}ε

where, as explained below, χ_{ℓ} is the ℓth cyclotomic character and ε is the Dirichlet character associated to f.

There is an incredible amount of “abuse of notation” packed into the above statement. Let M = \overline{Q}^{ker(ρ_{ℓ,f})} be the field fixed by the kernel of ρ_{ℓ,f}. Then the Frobenius element Frob_{P} (note P not p) is well defined as an element of Gal(M/Q), and the element Frob_{p} is then only well defined up to conjugacy. But this works out since ρ_{ℓ,f} is well-defined on Gal(M/Q) (it kills Gal(\overline{Q}/M)) and trace is welldefined on conjugacy classes (tr(AB) = tr(BA) so tr(ABA^{−1}) = T r(B)).

2.2 Characters

Let f ∈ S_{k}(N) be an eigenform for all Hecke operators. Then for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) with c ≡ 0 mod N we have

f (\frac{az + b}{cz + d}) = (cz + d)^{k}ε(d)f(z),

where ε : (Z/NZ)^{∗} → C^{∗} is a Dirichlet character mod N. If f is also normalized so that a_{1} = 1, as in Section 1.4, then ε actually takes values in K^{∗}.

Let G = Gal(\overline{Q}/Q), and let φ_{N} be the mod N cyclotomic character so that φ_{N} : G → (Z/NZ)^{∗} takes g ∈ G to the automorphism induced by g on the Nth cyclotomic extension Q(μ_{N}) of Q (where we identify Gal(Q(μ_{N})/Q) with (Z/NZ)^{∗}). Then what we called ε above in the formula det(ρ_{ℓ}) = χ_{ℓ}^{k−1}ε is really the composition

G ⟶^{φ_{N}} (Z/NZ)^{∗} ⟶^{ε} C^{∗}.

For each positive integer ν we consider the ℓ^{ν}th cyclotomic character on G,

φ_{ℓ^{ν}} : G → (Z/ℓ^{ν}Z)^{∗}.

Putting these together gives the ℓ-adic cyclotomic character

χ_{ℓ} : G → Z_{ℓ}^{∗}.

2.3 Parity conditions

Let c ∈ Gal(\overline{Q}/Q) be complex conjugation. Then φ_{N}(c) = −1 so ε(c) = ε(−1) and χ_{ℓ}^{k−1}(c) = (−1)^{k−1}. Letting ( \begin{matrix}a & b \\ c′ & d \end{matrix} ) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ), for f ∈ S_{k}(N), we have

f(z) = (−1)^{k}ε(−1)f(z),

so (−1)^{k}ε(−1) = 1. Thus

det(ρ_{ℓ,f}(c)) = ε(−1)(−1)^{k−1} = −1.

We say a representation is odd if the determinant of complex conjugation is −1. Thus the representation ρ_{ℓ,f} is odd. Remark 2.3.1 (Vague Question). How can one recognize representations like ρ_{ℓ,f} “in nature”? Fontaine and Mazur have made relevant conjectures. The modularity theorem can be reformulated by saying that for any representation ρ_{ℓ,E} coming from an elliptic curve E there is an f so that ρ_{ℓ,E} ≅ ρ_{ℓ,f}.

2.4 Conjectures of Serre (mod ℓ version)

Suppose f is a modular form, ℓ ∈ Z prime, λ a prime lying over ℓ, and the representation

ρ_{λ,f} : G → GL_{2}(K_{λ})

(constructed by Serre-Deligne) is irreducible. Then ρ_{λ,f} is conjugate to a representation with image in GL_{2}(O_{λ}), where O_{λ} is the ring of integers of K_{λ} (see Section 2.5 below). Reducing mod λ gives a representation

\overline{ρ}_{λ,f} : G → GL_{2}(F_{λ})

which has a well-defined trace and det, i.e., the det and trace do not depend on the choice of conjugate representation used to obtain the reduced representation. One knows from representation theory (the Brauer-Nesbitt theorem – see [CR62]) that if such a representation is semisimple then it is completely determined by its trace and det (more precisely, it is determined by the characteristic polynomials of all of its elements). Thus if \overline{ρ}_{λ,f} is irreducible (and hence semisimple) then it is unique in the sense that it does not depend on the choice of conjugate.

2.5 General remarks on mod p Galois representations

First, what are semisimple and irreducible representations? Remember that a representation ρ is a map from a group G to the endomorphisms of some vector space W (or a free module M if we are working over a ring instead of a field, but let’s not worry about that for now). A subspace W′ of W is said to be invariant under ρ if ρ takes W′ back into itself. (The point is that if W′ is invariant, then ρ induces representations on both W′ and W/W′.) An irreducible representation is one whose only invariant subspaces are {0} and W. A semisimple representation is one where for every invariant subspace W′ there is a complementary invariant subspace W′′ – that is, you can write ρ as the direct sum of ρ|_{W′} and ρ|_{W′′}.

Another way to say this is that if W′ is an invariant subspace then we get a short exact sequence

0 → ρ|_{W/W′} → ρ → ρ|_{W′} → 0.

Furthermore ρ is semisimple if and only if every such sequence splits.

Note that irreducible representations are semisimple. As mentioned above, twodimensional semisimple Galois representations are uniquely determined (up to isomorphism class) by their trace and determinant. In the case we are considering, G = Gal(\overline{Q}/Q) is compact, so the image of any Galois representation ρ into GL_{2}(K_{λ}) is compact. Thus we can conjugate it into GL_{2}(O_{λ}). Irreducibility is not needed for this.

Now that we have a representation into GL_{2}(O_{λ}), we can reduce to get a representation \overline{ρ} to GL_{2}(F_{λ}). This reduced representation is not uniquely determined by ρ, since we made a choice of basis (via conjugation) so that ρ would have image in GL_{2}(O_{λ}), and a different choice may lead to a non-isomorphic representation mod λ. However, the trace and determinant of a matrix are invariant under conjugation, so the trace and determinant of the reduced representation \overline{ρ} are uniquely determined by ρ.

So we know the trace and determinant of the reduced representation. If we also knew that it was semisimple, then we would know its isomorphism class, and we would be done. So we would be happy if the reduced representation is irreducible. And in fact, it is easy to see that if the reduced representation is irreducible, then ρ must also be irreducible. Most ρ of interest to us in this book will be irreducible. Unfortunately, the opposite implication does not hold: ρ irreducible need not imply that \overline{ρ} is irreducible.

2.6 Serre’s conjecture

Serre has made the following conjecture which is now a theorem (see [KW08]).

Conjecture 2.6.1 (Serre). All 2-dimensional irreducible representation of G over a finite field which are odd, i.e., det(σ(c)) = −1, c complex conjugation, are of the form \overline{ρ}_{λ,f} for some representation ρ_{λ,f} constructed as above.

Example 2.6.2. Let E/Q be an elliptic curve and let σ_{ℓ} : G → GL_{2}(F_{ℓ}) be the representation induced by the action of G on the ℓ-torsion of E. Then det σ_{ℓ} = φ_{ℓ} is odd and σ_{ℓ} is usually irreducible, so Serre’s conjecture implies that σ_{ℓ} is modular. From this one can, assuming Serre’s conjecture, prove that E is itself modular (see [Rib92]).

Definition 2.6.3 (Modular representation). Let σ : G → GL_{2}(F) (F is a finite field) be an irreducible representation of the Galois group G. Then we say that the representation σ is

Definition 2.6.4. modular if there is a modular form f, a prime λ, and an embedding F ↪ \overline{F}_{λ} such that σ ≅ \overline{ρ}_{λ,f} over \overline{F}_{λ}.

For more details, see Chapter 21 and [RS01].

2.7 Wiles’s perspective

Suppose E/Q is an elliptic curve and ρ_{ℓ,E} : G → GL_{2}(Z_{ℓ}) the associated ℓadic representation on the Tate module T_{ℓ}. Then by reducing we obtain a mod ℓ representation

\overline{ρ}_{ℓ,E} = σ_{ℓ,E} : G → GL_{2}(F_{ℓ}).

If we can show this representation is modular for infinitely many ℓ then we will know that E is modular.

Theorem 2.7.1 (Langlands and Tunnel). If σ_{2,E} and σ_{3,E} are irreducible, then they are modular.

This is proved by using that GL_{2}(F_{2}) and GL_{2}(F_{3}) are solvable so we may apply something called “base change for GL_{2}.”

Theorem 2.7.2 (Wiles). If ρ is an ℓ-adic representation which is irreducible and modular mod ℓ with ℓ > 2 and certain other reasonable hypothesis are satisfied, then ρ itself is modular.

12 2. Modular Representations and Algebraic Curves

Modular Forms of Level 1

In this chapter, we view modular forms of level 1 both as holomorphic functions on the upper half plane and functions on lattices. We then define Hecke operators on modular forms, and derive explicit formulas for the action of Hecke operators on q-expansions. An excellent reference for the theory of modular forms of level 1 is Serre [Ser73, Ch. 7].

3.1 The Definition

Let k be an integer. The space S_{k} = S_{k}(1) of cusp forms of level 1 and weight k consists of all functions f that are holomorphic on the upper half plane 𝔥 and such that for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) one has

f (\frac{aτ + b}{cτ + d}) = (cτ + d)^{k}f(τ),  (3.1.1)

and f vanishes at infinity, in a sense which we will now make precise. The matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) is in SL_{2}(Z), so f(τ + 1) = f(τ). Thus f passes to a well-defined function of q(τ) = e^{2πiτ}. Since for τ ∈ 𝔥 we have |q(τ)| < 1, we may view f(z) = F (q) as a function of q on the punctured open unit disc {q : 0 < |q| < 1}. The condition that f(τ) vanishes at infinity means that F (q) extends to a holomorphic function on the open disc {q : |q| < 1} so that F (0) = 0. Because holomorphic functions are represented by power series, there is a neighborhood of 0 such that

f(q) = ∑_{n=1}^{∞} a_{n}q^{n},

so for all τ ∈ 𝔥 with sufficiently large imaginary part (but see Remark 3.1.1 below), f(τ) = ∑_{n=1}^{∞} a_{n}e^{2πinτ}.

It will also be useful to consider the slightly larger space M_{k}(1) of holomorphic functions on 𝔥 that transform as above and are merely required to be holomorphic at infinity. Remark 3.1.1. In fact, the series ∑_{n=1}^{∞} a_{n}e^{2πinτ} converges for all τ ∈ 𝔥. This is because the Fourier coefficients a_{n} are O(n^{k/2}) (see [Miy89, Cor. 2.1.6, pg. 43]). Remark 3.1.2. In [Ser73, Ch. 7], the weight is defined in the same way, but in the notation our k is twice his k.

3.2 Some examples and conjectures

The space S_{k}(1) of cusp forms is a finite-dimensional complex vector space. For k even we have dim S_{k}(1) = ⌊k/12⌋ if k ≢ 2 (mod 12) and ⌊k/12⌋ − 1 if k ≡ 2 (mod 12), except when k = 2 in which case the dimension is 0. For even k, the space M_{k}(1) has dimension 1 more than the dimension of S_{k}(1), except when k = 2 when both have dimension 0. (For proofs, see, e.g., [Ser73, Ch. 7, §3].)

By the dimension formula mentioned above, the first interesting example is the space S_{12}(1), which is a 1-dimensional space spanned by

Δ(q) = q ∏_{n=1}^{∞}(1 − q^{n})^{24}
= q − 24q^{2} + 252q^{3} − 1472q^{4} + 4830q^{5} − 6048q^{6} − 16744q^{7} + 84480q^{8}

− 113643q^{9} − 115920q^{10} + 534612q^{11} − 370944q^{12} − 577738q^{13} + ⋯

That Δ lies in S_{12}(1) is proved in [Ser73, Ch. 7, §4.4] by expressing Δ in terms of elements of M_{4}(1) and M_{6}(1), and computing the q-expansion of the resulting expression. Example 3.2.1. We compute the q-expansion of Δ in Sage:

sage: delta_qexp(7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)

In Sage, computing delta_qexp(10^6) only takes a few seconds, and computing up to precision 10^8 is even reasonable. Sage does not use the formula q ∏_{n=1}^{∞}(1−q^{n})^{24} given above, which would take a very long time to directly evaluate, but instead uses the identity

Δ(q) = (∑_{n≥0}(−1)^{n}(2n + 1)q^{n(n+1)/2})^{8} ,

and computes the 8th power using asymptotically fast polynomial arithmetic in Z[q], which involves a discrete fast Fourier transform (implemented in [HH]).

The Ramanujan τ function τ(n) assigns to n the nth coefficient of Δ(q).

Conjecture 3.2.2 (Lehmer). τ(n) ≠ 0 for all n ≥ 1.

This conjecture has been verified for n ≤ 22798241520242687999 (Bosman, 2007

– see http://en.wikipedia.org/wiki/Tau-function).

Theorem 3.2.3 (Edixhoven et al.). Let p be a prime. There is a polynomial time algorithm to compute τ(p), polynomial in the number of digits of p.

Edixhoven’s idea is to use ℓ-adic cohomology and Arakelov theory to find an analogue of the Schoof-Elkies-Atkin algorithm (which counts the number N_{q} of points on an elliptic curves over a finite field F_{q} by computing N_{q} mod ℓ for many primes ℓ). Here’s some of what Edixhoven has to say about his result:

“You need to compute on varying curves such as X_{1}(ℓ) for ℓ up to log(p) say. An important by-product of my method is the computation of the mod ℓ Galois representations associated to Δ in time polynomial in ℓ. So, it should be seen as an attempt to make the Langlands correspondence for GL_{2} over Q available computationally.”

If f ∈ M_{k}(1) and g ∈ M_{k′}(1), then it is easy to see from the definitions that fg ∈ M_{k+k′}(1). Moreover, ⨁_{k≥0} M_{k}(1) is a commutative graded ring generated freely by E_{4} = 1+240 ∑_{n=1}^{∞} σ_{3}(n)q^{n} and E_{6} = 1−504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}, where σ_{d}(n) is the sum of the dth powers of the positive divisors of n (see [Ser73, Ch.7, §3.2]). Example 3.2.4. Because E_{4} and E_{6} generate, it is straightforward to write down a basis for any space M_{k}(1). For example, the space M_{36}(1) has basis

f_{1} = 1 + 6218175600q^{4} + 15281788354560q^{5} + ⋯
f_{2} = q + 57093088q^{4} + 37927345230q^{5} + ⋯
f_{3} = q^{2} + 194184q^{4} + 7442432q^{5} + ⋯
f_{4} = q^{3} − 72q^{4} + 2484q^{5} + ⋯

3.3 Modular forms as functions on lattices

In order to define Hecke operators, it will be useful to view modular forms as functions on lattices in C.

Definition 3.3.1 (Lattice). A lattice L ⊂ C is a subgroup L = Zω_{1} + Zω_{2} for which ω_{1}, ω_{2} ∈ C are linearly independent over R.

We may assume that ω_{1}/ω_{2} ∈ 𝔥 = {z ∈ C : Im(z) > 0}. Let R be the set of all lattices in C. Let E be the set of isomorphism classes of pairs (E, ω), where E is an elliptic curve over C and ω ∈ Ω_{E}^{1} is a nonzero holomorphic differential 1-form on E. Two pairs (E, ω) and (E′, ω′) are isomorphic if there is an isomorphism φ : E → E′ such that φ^{∗}(ω′) = ω.

Proposition 3.3.2. There is a bijection between R and E under which L ∈ R corresponds to (C/L, dz) ∈ E.

Proof. We describe the maps in each direction, but leave the proof that they induce a well-defined bijection as an exercise for the reader [[add ref to actual exercise]]. Given L ∈ R, by Weierstrass theory the quotient C/L is an elliptic curve, which is equipped with the distinguished differential ω induced by the differential dz

on C. [[TODO: See [Kat73, Appendix A1.1] where the exact curve, weierstrass ℘ and that ω = dx/y corresponds to dz is explained nicely.]]

Conversely, if E is an elliptic curve over C and ω ∈ Ω_{E}^{1} is a nonzero differential, we obtain a lattice L in C by integrating homology classes:

L = L_{ω} = {∫_{γ} ω : γ ∈ H_{1}(E(C), Z)} .

Let

B = {(ω_{1}, ω_{2}) : ω_{1}, ω_{2} ∈ C, ω_{1}/ω_{2} ∈ 𝔥} ,

be the set of ordered basis of lattices in C, ordered so that ω_{1}/ω_{2} ∈ 𝔥. There is a left action of SL_{2}(Z) on B given by

( \begin{matrix}a & b \\ c & d \end{matrix} ) (ω_{1}, ω_{2}) ↦ (aω_{1} + bω_{2}, cω_{1} + dω_{2})

and SL_{2}(Z)\B ≅ R. (The action is just the left action of matrices on column vectors, except we write (ω_{1}, ω_{2}) as a row vector since it takes less space.)

Give a modular form f ∈ M_{k}(1), associate to f a function F : R → C as follows. First, on lattices of the special form Zτ + Z, for τ ∈ 𝔥, let F (Zτ + Z) = f(τ).

In order to extend F to a function on all lattices, note that F satisfies the homogeneity condition F (λL) = λ^{−k}F (L), for any λ ∈ C and L ∈ R. Then

F (Zω_{1} + Zω_{2}) = ω_{2}^{−k}F (Zω_{1}/ω_{2} + Z) := ω_{2}^{−k}f(ω_{1}/ω_{2}).

That F is well-defined exactly amounts to the transformation condition (3.1.1) that f satisfies.

Lemma 3.3.3. The lattice function F : R → C associated to f ∈ M_{k}(1) is well defined.

Proof. Suppose Zω_{1} + Zω_{2} = Zω′_{1} + Zω′_{2} with ω_{1}/ω_{2} and ω′_{1}/ω′_{2} both in 𝔥. We must verify that ω_{2}^{−k}f(ω_{1}/ω_{2}) = (ω′_{2})^{−k}f(ω′_{1}/ω′_{2}). There exists ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) such that ω′_{1} = aω_{1} + bω_{2} and ω′_{2} = cω_{1} + dω_{2}. Dividing, we see that ω′_{1}/ω′_{2} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) (ω_{1}/ω_{2}). Because f is a weight k modular form, we have

f (\frac{ω′_{1}}{ω′_{2}} ) = f ((\begin{matrix}a & b \\ c & d \end{matrix}) (\frac{ω_{1}}{ω_{2}} )) = (c\frac{ω_{1}}{ω_{2}} + d)^{k} f (\frac{ω_{1}}{ω_{2}} ) .

Multiplying both sides by ω_{2}^{k} yields

ω_{2}^{k}f (\frac{ω′_{1}}{ω′_{2}} ) = (cω_{1} + dω_{2})^{k}f (\frac{ω_{1}}{ω_{2}} ) .

Observing that ω′_{2} = cω_{1} + dω_{2} and dividing again completes the proof.

Since f(τ) = F (Zτ +Z), we can recover f from F, so the map f ↦ F is injective. Moreover, it is surjective in the sense that if F is homogeneous of degree −k, then F arises from a function f : 𝔥 → C that transforms like a modular form. More precisely, if F : R → C satisfies the homogeneity condition F (λL) = λ^{−k}F (L), then the function f : 𝔥 → C defined by f(τ) = F (Zτ +Z) transforms like a modular form of weight k, as the following computation shows: For any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) and τ ∈ 𝔥, we have

f (\frac{aτ + b}{cτ + d}) = F (Z\frac{aτ + b}{cτ + d} + Z)
= F ((cτ + d)^{−1} (Z(aτ + b) + Z(cτ + d)))
= (cτ + d)^{k}F (Z(aτ + b) + Z(cτ + d))
= (cτ + d)^{k}F (Zτ + Z)
= (cτ + d)^{k}f(τ).

Say that a function F : R → C is holomorphic on 𝔥 ∪ {∞} if the function f(τ) = F (Zτ + Z) is. We summarize the above discussion in a proposition.

Proposition 3.3.4. There is a bijection between M_{k}(1) and functions F : R → C that are homogeneous of degree −k and holomorphic on 𝔥 ∪ {∞}. Under this bijection F : R → C corresponds to f(τ) = F (Zτ + Z).

3.4 Hecke operators

Define a map T_{n} from the free abelian group generated by all C-lattices into itself by

T_{n}(L) = ∑_{\begin{matrix}L′⊂L \\ [L:L′]=n \end{matrix}} L′,

where the sum is over all sublattices L′ ⊂ L of index n. For any function F : R → C on lattices, define T_{n}(F) : R → C by

(T_{n}(F))(L) = n^{k−1} ∑_{\begin{matrix}L′⊂L \\ [L:L′]=n \end{matrix}} F (L′).

If F is homogeneous of degree −k, then T_{n}(F) is also homogeneous of degree −k.

In the next Section, we will show that gcd(n, m) = 1 implies T_{n}T_{m} = T_{nm} and that T_{p^{k}} is a polynomial in Z[T_{p}].

Suppose L′ ⊂ L with [L : L′] = n. Then every element of L/L′ has order dividing n, so nL ⊂ L′ ⊂ L and

L′/nL ⊂ L/nL ≈ (Z/nZ)^{2}.

Thus the subgroups of (Z/nZ)^{2} of order n correspond to the sublattices L′ of L of index n. When n = ℓ is prime, there are ℓ + 1 such subgroups, since the subgroups correspond to nonzero vectors in F_{ℓ} modulo scalar equivalence, and there are (ℓ^{2} − 1)/(ℓ − 1) = ℓ + 1 of them.

Recall from Proposition 3.3.2 that there is a bijection between the set R of lattices in C and the set E of isomorphism classes of pairs (E, ω), where E is an elliptic curve over C and ω is a nonzero differential on E.

Suppose F : R → C is homogeneous of degree −k, so F (λL) = λ^{−k}F (L). Then we may also view T_{ℓ} as a sum over lattices that contain L with index ℓ, as follows.

Suppose L′ ⊂ L is a sublattice of index ℓ and set L′′ = ℓ^{−1}L′. Then we have a chain of inclusions

ℓL ⊂ L′ ⊂ L ⊂ ℓ^{−1}L′ = L′′.

Since [ℓ^{−1}L′ : L′] = ℓ^{2} and [L : L′] = ℓ, it follows that [L′′ : L] = ℓ. Because F is homogeneous of degree −k,

T_{ℓ}(F)(L) = ℓ^{k−1} ∑_{[L:L′]=ℓ} F (L′) = \frac{1}{ℓ} ∑_{[L′′:L]=ℓ} F (L′′).  (3.4.1)

3.4.1 Relations Between Hecke Operators

In this section we show that the Hecke operators T_{n}, viewed as functions on the free abelian group on lattices via

T_{n}(L) = ∑_{\begin{matrix}L′⊂L \\ [L:L′]=n \end{matrix}} L′

are multiplicative satisfy a recurrence for prime powers. Let R_{p}(L) = pL ∈ R be the lattice L scaled by p (this is not p copies of L in the free abelian group).

Proposition 3.4.1. If gcd(n, m) = 1, then

T_{nm} = T_{n}T_{m}.  (3.4.2)

If r ≥ 1 and p is a prime, then

T_{p^{r+1}} = T_{p^{r}}T_{p} − pR_{p}T_{p^{r−1}}.  (3.4.3)

Proof. (Compare [Ser73, Cor. 1, pg. 99].)

Proving relation (3.4.2) is equivalent to showing that for every sublattice L′′ of L of index nm, there is a unique sublattice L′ ⊂ L with L′′ ⊂ L′ such that [L : L′] = m and [L′ : L′′] = n. To see this, note that the abelian group L/L′′ is of order nm, so it decomposes uniquely as a product of subgroups of orders n and m. The unique subgroup of L/L′′ of order m corresponds to the unique sublattice L′ ⊂ L such that [L : L′] = m and [L′ : L′′] = n.

To prove (3.4.3), let L be a lattice and note that

T_{p^{r}}T_{p}(L),  T_{p^{r+1}}(L),  and  R_{p}T_{p^{r−1}}(L)

are all linear combinations of lattices of index p^{r+1} in L (note that R_{p} commutes with T_{p^{r−1}} and [L : R_{p}(L)] = p^{2}). Let L′′ be a lattice of index p^{r+1} in L. In the linear combination L′′ appears with coefficients a, b, c ∈ Z (say), and our goal is to prove that a = b + pc. Note that b = 1, since the lattices in the sum T_{p^{r+1}}(L) each appear exactly once, so in fact we must prove that a = 1 + pc. The lattices appearing in R_{p}T_{p^{r−1}}(L) = T_{p^{r−1}}(R_{p}L) = T_{p^{r−1}}(pL) are exactly those of index p^{r−1} in pL, each with multiplicity 1. We consider two cases, depending on whether or not L′′ is in that sum.

• Case L′′ ⊂ pL, so c = 1: We must show that a = 1 + p. Every single one of the p + 1 sublattices of L of index p must contain pL, so they also all contain L′′ ⊂ pL. Thus a = p + 1, as claimed.

3.5 Hecke operators directly on q-expansions 19

• Case L′′ ⊄ pL, so c = 0: We must show that a = 1. We have that a is the number of lattices L′ of index p in L with L′′ ⊂ L′ ⊂ L. Let L′ be such a lattice. Since [L : L′] = p, we have pL ⊂ L′, and the image of L′ in L/pL is of order p. Since L′′ ⊄ pL, the image of L′′ in L/pL is also of order p (the image is nonzero and contained in the image of L′), and since L′′ ⊂ L′, we thus must have that the images of L′′ and L′ in L/pL are the same subgroup of order p. Hence L′ is completely determined by L′′, so there is exactly one L′ that works, hence a = 1.

Corollary 3.4.2. The Hecke operators T_{n} commute with each other, for all n.

3.5 Hecke operators directly on q-expansions

Recall that the nth Hecke operator T_{n} of weight k on lattice functions is given by

T_{n}(F)(L) = n^{k−1} ∑_{\begin{matrix}L′⊂L \\ [L:L′]=n \end{matrix}} F (L′).  (3.5.1)

Modular forms of weight k correspond to holomorphic functions of degree −k on lattices, and each T_{n} extends to an operator on these functions on lattices, so T_{n} defines on operator on M_{k}(1).

Extending F linearly to a function on the free abelian group on lattices, we have

T_{n}(F)(L) = n^{k−1}F (T_{n}(L)),

which allows us to apply Proposition 3.4.1.

Proposition 3.5.1. The above action of the Hecke operators on homomogenous lattice function F : R → C of degree −k (equivalently, on M_{k}(1)) satisfies the following relations:

T_{nm} = T_{n}T_{m}  if gcd(n, m) = 1,
T_{p^{r+1}} = T_{p^{r}}T_{p} − p^{k−1}T_{p^{r−1}}  if r ≥ 1 and p prime.

Proof. We compute F of both sides of the formulas in Proposition 3.4.1, applied to a lattice L. The first relation is immediate since (nm)^{k−1} = n^{k−1}m^{k−1}. For the second, note that having extended F linearly to the free abelian group on lattices, we have

F ((pR_{p})(L)) = p · F (pL) = p · p^{−k}F (L) = p^{1−k}F (L).

Thus (3.4.3) implies that if L is a lattice, then

F (T_{p^{r+1}}(L)) = F (T_{p^{r}}T_{p}(L)) − p^{1−k}F (T_{p^{r−1}}(L)).  (3.5.2)

Unwinding definitions, our goal is to prove that

(p^{r+1})^{k−1}F (T_{p^{r+1}}(L)) = (p^{r})^{k−1}p^{k−1}F (T_{p^{r}}T_{p}(L)) − p^{k−1}(p^{r−1})^{k−1}F (T_{p^{r−1}}(L)).

Dividing both sides by p^{(r+1)(k−1)} and using that (3.5.2) holds, we see that this follows from the fact that

\frac{p^{k−1}(p^{r−1})^{k−1}}{p^{(r+1)(k−1)}} = p^{1−k}.

A holomorphic function on the unit disk is determined by its Fourier expansion, so Fourier expansion defines an injective map M_{k}(1) ↪ C[[q]]. In this section, we describe T_{n}(∑ a_{m}q^{m}) explicitly as a q-expansion.

3.5.1 Explicit description of sublattices

In order to describe T_{n} more explicitly, we enumerate the sublattices L′ ⊂ L of index n. More precisely, we give a basis for each L′ in terms of a basis for L. Note that L/L′ is a group of order n and

L′/nL ⊂ L/nL = (Z/nZ)^{2}.

Write L = Zω_{1} + Zω_{2}, let Y_{2} be the cyclic subgroup of L/L′ generated by ω_{2} and let d = #Y_{2}. If Y_{1} = (L/L′)/Y_{2}, then Y_{1} is generated by the image of ω_{1}, so it is a cyclic group of order a = n/d. Our goal is to exhibit a basis of L′. Let ω′_{2} = dω_{2} ∈ L′ and use that Y_{1} is generated by the image of ω_{1} to write aω_{1} = ω′_{1} − bω_{2} for some integer b and some ω′_{1} ∈ L′. Since b is only well-defined modulo d we may assume 0 ≤ b ≤ d − 1. Thus

(\begin{matrix}ω′_{1} \\ ω′_{2} \end{matrix}) = (\begin{matrix}a & b \\ 0 & d \end{matrix})(\begin{matrix}ω_{1} \\ ω_{2} \end{matrix})

and the change of basis matrix has determinant ad = n. Since

Zω′_{1} + Zω′_{2} ⊂ L′ ⊂ L = Zω_{1} + Zω_{2}

and [L : Zω′_{1} + Zω′_{2}] = n (since the change of basis matrix has determinant n) and [L : L′] = n we see that L′ = Zω′_{1} + Zω′_{2}.

Proposition 3.5.2. Let n be a positive integer. There is a one-to-one correspondence between sublattices L′ ⊂ L of index n and matrices ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) with ad = n and 0 ≤ b ≤ d − 1.

Proof. The correspondence is described above. To check that it is a bijection, we just need to show that if γ = ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) and γ′ = ( \begin{matrix}a′ & b′ \\ 0 & d′ \end{matrix} ) are two matrices satisfying the listed conditions, and

Z(aω_{1} + bω_{2}) + Zdω_{2} = Z(aω′_{1} + bω′_{2}) + Zdω′_{2},

then γ = γ′. Equivalently, if σ ∈ SL_{2}(Z) and σγ = γ′, then σ = 1. To see this, we compute

σ = γ′γ^{−1} = \frac{1}{n} (\begin{matrix}a′d & ab′ − a′b \\ 0 & ad′ \end{matrix}) .

Since σ ∈ SL_{2}(Z), we have n | a′d, and n | ad′, and aa′dd′ = n^{2}. If a′d > n, then because aa′dd′ = n^{2}, we would have ad′ < n, which would contradict the fact that n | ad′; also, a′d < n is impossible since n | a′d. Thus a′d = n and likewise ad′ = n. Since ad = n as well, it follows that a′ = a and d′ = d, so σ = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) for some t ∈ Z. Then σγ = ( \begin{matrix}a & b+dt \\ 0 & d \end{matrix} ), which implies that t = 0, since 0 ≤ b ≤ d − 1 and 0 ≤ b + dt ≤ d − 1.

Remark 3.5.3. As mentioned earlier, when n = ℓ is prime, there are ℓ+1 sublattices of index ℓ. In general, the number of such sublattices is the sum of the positive divisors of n (exercise)^{1} . 1

3.5.2 Hecke operators on q-expansions

Recall that if f ∈ M_{k}(1), then f is a holomorphic function on 𝔥 ∪ {∞} such that

f(τ) = f (\frac{aτ + b}{cτ + d}) (cτ + d)^{−k}

for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z). Using Fourier expansion we write

f(τ) = ∑_{m=0}^{∞} c_{m}e^{2πiτm},

and say f is a cusp form if c_{0} = 0. Also, there is a bijection between modular forms f of weight k and holomorphic lattice functions F : R → C that satisfy F (λL) = λ^{−k}F (L); under this bijection F corresponds to f(τ) = F (Zτ + Z).

Now assume f(τ) = ∑_{m=0}^{∞} c_{m}q^{m} is a modular form with corresponding lattice function F. Using the explicit description of sublattices from Section 3.5.1 above, we can describe the action of the Hecke operator T_{n} on the Fourier expansion of f(τ), as follows:

T_{n}F (Zτ + Z) = n^{k−1} ∑_{\begin{matrix}a,b,d \\ ad=n \\ 0≤b≤d−1 \end{matrix}} F ((aτ + b)Z + dZ)

= n^{k−1} ∑ d^{−k}F (\frac{aτ + b}{d}Z + Z)

= n^{k−1} ∑ d^{−k}f (\frac{aτ + b}{d})

= n^{k−1} ∑_{a,d,b,m} d^{−k}c_{m}e^{2πi(\frac{aτ+b}{d})m}

= n^{k−1} ∑_{a,d,m} d^{1−k}c_{m}e^{\frac{2πiamτ}{d}} \frac{1}{d} ∑_{b=0}^{d−1} (e^{\frac{2πim}{d}} )^{b}

= n^{k−1} ∑_{\begin{matrix}ad=n \\ m′≥0 \end{matrix}} d^{1−k}c_{dm′}e^{2πiam′τ}

= ∑_{\begin{matrix}ad=n \\ m′≥0 \end{matrix}} a^{k−1}c_{dm′}q^{am′}.

1Put reference to actual exercise

In the second to the last expression we let m = dm′ for m′ ≥ 0, then use that the sum \frac{1}{d} ∑_{b=0}^{d−1}(e^{\frac{2πim}{d}})^{b} is only nonzero if d | m, in which case the sum equals 1.

Thus

T_{n}f(q) = ∑_{\begin{matrix}ad=n \\ m≥0 \end{matrix}} a^{k−1}c_{dm}q^{am}.

Put another way, if μ is a nonnegative integer, then the coefficient of q^{μ} is

∑_{\begin{matrix}a|n \\ a|μ \end{matrix}} a^{k−1}c_{\frac{nμ}{a^{2}}} .

(To see this, let m = μ/a and d = n/a and substitute into the formula above.) Remark 3.5.4. When k ≥ 1 the coefficients of q^{μ} for all μ belong to the Z-module generated by the c_{m}. Remark 3.5.5. Setting μ = 0 gives the constant coefficient of T_{n}f which is

∑_{a|n} a^{k−1}c_{0} = σ_{k−1}(n)c_{0}.

Thus if f is a cusp form so is T_{n}f. (T_{n}f is holomorphic since its original definition is as a finite sum of holomorphic functions.) Remark 3.5.6. Setting μ = 1 shows that the coefficient of q in T_{n}f is ∑_{a|1} a^{k−1}c_{n} = c_{n}. As an immediate corollary we have the following important result.

Corollary 3.5.7. If f is a cusp form such that T_{n}f has 0 as coefficient of q for all n ≥ 1, then f = 0.

In the special case when n = p is prime, the action action of T_{p} on the qexpansion of f is given by the following formula:

T_{p}f = ∑_{μ≥0} ∑_{\begin{matrix}a|p \\ a|μ \end{matrix}} a^{k−1}c_{\frac{nμ}{a^{2}}} q^{μ}.

Since p is prime, either a = 1 or a = p. When a = 1, c_{pμ} occurs in the coefficient of q^{μ} and when a = p, we can write μ = pλ and we get terms p^{k−1}c_{λ} in q^{pλ}. Thus

T_{p}f = ∑_{μ≥0} c_{pμ}q^{μ} + p^{k−1} ∑_{λ≥0} c_{λ}q^{pλ}.

3.5.3 The Hecke algebra and eigenforms

Definition 3.5.8 (Hecke Algebra). The Hecke algebra T associated to M_{k}(1) is the subring of End(M_{k}(1)) generated by the operators T_{n} for all n. Similarly, the Hecke algebra associated to S_{k}(1) is the subring of End(S_{k}(1)) generated by all Hecke operators T_{n}.

The Hecke algebra is commutative because T_{p^{ν}} is a polynomial in T_{p} and when gcd(n, m) = 1 we have T_{n}T_{m} = T_{nm} = T_{mn} = T_{m}T_{n} (see Section 3.4.1). Also, T is of finite rank over Z, because of Remark 3.5.4 and that the finite dimensional space S_{k}(1) has a basis with q-expansions in Z[[q]].

Definition 3.5.9 (Eigenform). An eigenform f ∈ M_{k}(1) is a nonzero element such that f is an eigenvector for every Hecke operator T_{n}. If f ∈ S_{k}(1) is an eigenform, then f is normalized if the coefficient of q in the q-expansion of f is 1. We sometimes called a normalized cuspidal eigenform a newform.

If f = ∑_{n=1}^{∞} c_{n}q^{n} is a normalized eigenform, then Remark 3.5.6 implies that T_{n}(f) = c_{n}f. Thus the coefficients of a newform are exactly the system of eigenvalues of the Hecke operators acting on the newform.

Remark 3.5.10. It follows from Victor Miller’s thesis [[ref my modform book??]] that T_{1}, …, T_{n} generate T ⊂ End(S_{k}(1)), where n = dim S_{k}(1).

3.5.4 Examples

We compute the space of weight 12 modular forms of level 1, along with its cuspidal subspace:

sage: M = ModularForms(1,12, prec=3)
sage: M.basis()

[

q - 24*q^2 + O(q^3),
1 + 65520/691*q + 134250480/691*q^2 + O(q^3)

]

sage: M.hecke_matrix(2)
[  -24     0]
[    0 2049]
sage: S = M.cuspidal_subspace()
sage: S.hecke_matrix(2)

[-24]

sage: factor(M.hecke_polynomial(2))
(x - 2049) * (x + 24)

We also compute the space of forms of weight 40:

sage: M = ModularForms(1,40)
sage: M.basis()

[

q + 19291168*q^4 + 37956369150*q^5 + O(q^6),
q^2 + 156024*q^4 + 57085952*q^5 + O(q^6),
q^3 + 168*q^4 - 12636*q^5 + O(q^6),
1 + 1082400/261082718496449122051*q + ...

]

sage: M.hecke_matrix(2)
[                0    549775105056  14446985236992                 0]
[                1           156024      1914094476                 0]
[                0               168           392832                 0]
[                0                 0                 0    549755813889]
sage: factor(M.hecke_polynomial(2))
(x - 549755813889) *
      (x^3 - 548856*x^2 - 810051757056*x + 213542160549543936)

3.6 Two Conjectures about Hecke operators on level 1 modular forms

3.6.1 Maeda’s conjecture

Conjecture 3.6.1 (Maeda). Let k be a positive integer such that S_{k}(1) has positive dimension and let T ⊂ End(S_{k}(1)) be the Hecke algebra. Then there is only one Gal(\overline{Q}/Q) orbit of normalized eigenforms of level 1.

There is some numerical evidence for this conjecture. It is true for k ≤ 2000, according to [FJ02]. The MathSciNet reviewer of [FJ02] said “In the present paper the authors take a big step forward towards proving Maeda’s conjecture in the affirmative by establishing that the Hecke polynomial T_{p,k}(x) is irreducible and has full Galois group over ℚ for k ≤ 2000 and p < 2000, p prime.” Using Sage, Alex Ghitza verified the conjecture for k ≤ 4096 (see [Ghi]). Buzzard shows in

[Buz96] that for the weights k ≤ 228 with k/12 a prime, the Galois group of the characteristic polynomial of T_{2} is the full symmetric group, and is, in particular, irreducible.

3.6.2 The Gouvea-Mazur conjecture

Fix a prime p, and let F_{p,k} ∈ Z[x] be the characteristic polynomial of T_{p} acting on M_{k}(1). The slopes of F_{p,k} are the p-adic valuations ord_{p}(α) ∈ Q of the roots α ∈ \overline{Q}_{p} of F_{p,k}. They can be computed easily using Newton polygons.^{2} For example, the 2 p = 5 slopes for F_{5,12} are 0, 1, 1, for F_{5,12+4·5} they are 0, 1, 1, 4, 4, and for F_{5,12+4·5^{2}} they are 0, 1, 1, 5, 5, 5, 5, 5, 5, 10, 10, 11, 11, 14, 14, 15, 15, 16, 16.

sage: def s(k,p):
...         M = ModularForms(1,k)
...         v = M.hecke_polynomial(p).newton_slopes(p)
...         return  list(sorted(v))
sage: s(12,5)
[0, 1]
sage: s(12 + 4*5, 5)
[0, 1, 4]
sage: s(12 + 4*5^2, 5)
[0, 1, 5, 5, 5, 10, 11, 14, 15, 16]
sage: s(12 + 4*5^3, 5)            # long  time
!! WAY  TOO  SLOW  -- TODO  -- see  trac  9749 !!

Instead, we compute the slopes more directly as follows (this is fast):

sage: def s(k,p):
...         d = dimension_modular_forms(1, k)
...         B = victor_miller_basis(k, p*d+1)
...         T = hecke_operator_on_basis(B, p, k)
...         return  list(sorted(T.charpoly().newton_slopes(p)))
sage: s(12,5)

2Jared Weinstein suggests we add some background explaining newton polygons and why they are helpful.

[0, 1]
sage: s(12 + 4*5, 5)
[0, 1, 4]
sage: s(12 + 4*5^2, 5)
[0, 1, 5, 5, 5, 10, 11, 14, 15, 16]
sage: s(12 + 4*5^3, 5)
[0, 1, 5, 5, 5, 10, 11, 14, 15, 16, 20, 21, 24, 25, 27,
 30, 31, 34, 36, 37, 40, 41, 45, 46, 47, 50, 51, 55, 55,
 55, 59, 60, 63, 64, 65, 69, 70, 73, 74, 76, 79, 80, 83]

Let d(k, α, p) be the multiplicity of α as a slope of F_{p,k}.

Conjecture 3.6.2 (Gouvea-Mazur, 1992). Fix a prime p and a nonnegative rational number α. Suppose k_{1} and k_{2} are integers with k_{1}, k_{2} ≥ 2α + 2, and k_{1} ≡ k_{2} (mod p^{n}(p − 1)) for some integer n ≥ α. Then d(k_{1}, α, p) = d(k_{2}, α, p).

Notice that the above examples, with p = 5 and k_{1} = 12, are consistent with this conjecture. However, it came as a huge surprise that the conjecture is false in general!

Frank Calegari and Kevin Buzzard [BC04] found the first counterexample, when p = 59, k_{1} = 16, α = 1, and k_{2} = 16 + 59 · 58 = 3438. We have d(16, 0, 59) = 0, d(16, 1, 59) = 1, and d(16, α, 59) = 0 for all other α. However, initial computations strongly suggest (but do not prove!) that d(3438, 1, 59) = 2. It is a finite, but difficult, computation to decide what d(3438, 1, 59) really is (see Section 3.7). Using a trace formula, Calegari and Buzzard at least showed that either d(3438, 1, 59) ≥ 2 or there exists α < 1 such that d(3438, α, 59) > 0, both of which contradict Conjecture 3.6.2.

There are many theorems about more general formulations of the Gouvea-Mazur conjecture, and a whole geometric theory “the Eigencurve” [CM98] that helps explain it, but discussing this further is beyond the scope of this book.

3.7 An Algorithm for computing characteristic polynomials of Hecke operators

In computational investigations, it is frequently useful to compute the characteristic polynomial of the Hecke operator T_{p,k} of T_{p} acting on S_{k}(1). This can be accomplished in several ways, each of which has advantages. The Eichler-Selberg trace formula (see Zagier’s appendix to [Lan95, Ch. III]), can be used to compute the trace of T_{n,k}, for n = 1, p, p^{2}, …, p^{d−1}, where d = dim S_{k}(1), and from these traces it is straightforward to recover the characteristic polynomial of T_{p,k}. Using the trace formula, the time required to compute Tr(T_{n,k}) grows “very quickly” in n (though not in k), so this method becomes unsuitable when the dimension is large or p is large, since p^{d−1} is huge. Another alternative is to use modular symbols of weight k, as in [Mer94], but if one is only interested in characteristic polynomials, little is gained over more naive methods (modular symbols are most useful for investigating special values of L-functions).

In this section, we describe an algorithm to compute the characteristic polynomial of the Hecke operator T_{p,k}, which is adapted for the case when p > 2. It could be generalized to modular forms for Γ_{1}(N), given a method to compute a basis of q-expansions to “low precision” for the space of modular forms of weight k and level N. By “low precision” we mean to precision O(q^{dp+1}), where T_{1}, T_{2}, …, T_{d} generate the Hecke algebra T as a ring. The algorithm described here uses nothing more than the basics of modular forms and some linear algebra; in particular, no trace formulas or modular symbols are involved.

3.7.1 Review of basic facts about modular forms

We briefly recall the background for this section. Fix an even integer k. Let M_{k}(1) denote the space of weight k modular forms for SL_{2}(Z) and S_{k}(1) the subspace of cusp forms. Thus M_{k}(1) is a C-vector space that is equipped with a ring

T = Z[… T_{p,k} …] ⊂ End(M_{k}(1))

of Hecke operators. Moreover, there is an injective q-expansion map M_{k}(1) ↪ C[[q]]. For example, when k ≥ 4 there is an Eisenstein series E_{k}, which lies in M_{k}(1). The first two Eisenstein series are

E_{4}(q) = \frac{1}{240} + ∑_{n≥1} σ_{3}(n)q^{n} and E_{6}(q) = \frac{1}{504} + ∑_{n≥1} σ_{5}(n)q^{n},

where q = e^{2πiz}, σ_{k−1}(n) is the sum of the k − 1st power of the positive divisors. For every prime number p, the Hecke operator T_{p,k} acts on M_{k}(1) by

T_{p,k} (∑_{n≥0} a_{n}q^{n}) = ∑_{n≥0} a_{np}q^{n} + p^{k−1}a_{n}q^{np}.  (3.7.1)

Proposition 3.7.1. The set of modular forms E_{4}^{a}E_{6}^{b} is a basis for M_{k}(1), where a and b range through nonnegative integers such that 4a+6b = k. Moreover, S_{k}(1) is the subspace of M_{k}(1) of elements whose q-expansions have constant coefficient 0.

3.7.2 The Naive approach

Let k be an even positive integer and p be a prime. Our goal is to compute the characteristic polynomial of the Hecke operator T_{p,k} acting on S_{k}(1). In practice, when k and p are both reasonably large, e.g., k = 886 and p = 59, then the coefficients of the characteristic polynomial are huge (the roots of the characteristic polynomial are O(p^{k/2−1})). A naive way to compute the characteristic polynomial of T_{p,k} is to use (3.7.1) to compute the matrix [T_{p,k}] of T_{p,k} on the basis of Proposition 3.7.1, where E_{4} and E_{6} are computed to precision p dim M_{k}(1), and to then compute the characteristic polynomial of [T_{p,k}] using, e.g., a modular algorithm (compute the characteristic polynomial modulo many primes, and use the Chinese Remainder Theorem). The difficulty with this approach is that the coefficients of the q-expansions of E_{4}^{a}E_{6}^{b} to precision p dim M_{k}(1) quickly become enormous, so both storing them and computing with them is costly, and the components of [T_{p,k}] are also huge so the characteristic polynomial is difficult to compute. See Example 3.2.4 above, where the coefficients of the q-expansions are already large.

3.7.3 The Eigenform method

We now describe another approach to computing characteristic polynomials, which gets just the information required. Recall Maeda’s conjecture from Section 3.6.1, which asserts that S_{k}(1) is spanned by the Gal(\overline{Q}/Q)-conjugates of a single eigenform f = ∑ b_{n}q^{n}. For simplicity of exposition below, we assume this conjecture, though the algorithm can probably be modified to deal with the general case. We will refer to this eigenform f, which is well-defined up to Gal(\overline{Q}/Q)-conjugacy, as Maeda’s eigenform.

Lemma 3.7.2. The characteristic polynomial of the pth coefficient b_{p} of Maeda’s eigenform f, in the field Q(b_{1}, b_{2}, …), is equal to the characteristic polynomial of T_{p,k} acting on S_{k}(1).

Proof. The map T ⊗ Q → Q(b_{1}, b_{2}, …) that sends T_{n} → b_{n} is an isomorphism of Q-algebras.

Victor Miller shows in his thesis that S_{k}(1) has a unique basis f_{1}, …, f_{d} ∈ Z[[q]] with a_{i}(f_{j}) = δ_{ij}, i.e., the first d × d block of coefficients is the identity matrix. Again, in the general case, the requirement that there is such a basis can be avoided, but for simplicity of exposition we assume there is such a basis. We refer to the basis f_{1}, …, f_{d} as Miller’s basis.

Algorithm 3.7.3. We assume in the algorithm that the characteristic polynomial of T_{2} has no multiple roots (this is easy to check, and if false, then you have found an interesting counterexample to the conjecture that the characteristic polynomial of T_{2} has Galois group the full symmetric group).

1. Using Proposition 3.7.1 and Gauss elimination, we compute Miller’s basis f_{1}, …, f_{d} to precision O(q^{2d+1}), where d = dim S_{k}(1). This is exactly the precision needed to compute the matrix of T_{2}.

2. Using Definition 3.7.1, we compute the matrix [T_{2}] of T_{2} with respect to Miller’s basis f_{1}, …, f_{d}. We compute the matrix with respect to the Miller basis mainly because it makes the linear algebra much simpler.

3. Using Algorithm 3.7.5 below we write down an eigenvector e = (e_{1}, …, e_{d}) ∈ K^{d} for [T_{2}]. In practice, the components of T_{2} are not very large, so the numbers involved in computing e are also not very large.

4. Since e_{1}f_{1} +⋯+e_{d}f_{d} is an eigenvector for T_{2}, our assumption that the characteristic polynomial of T_{2} is square free (and the fact that T is commutative) implies that e_{1}f_{1} + ⋯ + e_{d}f_{d} is also an eigenvector for T_{p}. Normalizing, we see that up to Galois conjugacy,

b_{p} = ∑_{i=1}^{d} \frac{e_{i}}{e_{1}} · a_{p}(f_{i}),

where the b_{p} are the coefficients of Maeda’s eigenform f. For example, since the f_{i} are Miller’s basis, if p ≤ d then

b_{p} = \frac{e_{p}}{e_{1}} if p ≤ d,

since a_{p}(f_{i}) = 0 for all i ≠ p and a_{p}(f_{p}) = 1.

5. Finally, once we have computed b_{p}, we can compute the characteristic polynomial of T_{p}, because it is the minimal polynomial of b_{p}. We spend the rest of this section discussing how to make this step practical.

Computing b_{p} directly in step 4 is extremely costly because the divisions e_{i}/e_{1} lead to massive coefficient explosion, and the same remark applies to computing the minimal polynomial of b_{p}. Instead we compute the reductions \overline{b}_{p} modulo ℓ and the characteristic polynomial of \overline{b}_{p} modulo ℓ for many primes ℓ, then recover only the characteristic polynomial of b_{p} using the Chinese Remainder Theorem. Deligne’s bound on the magnitude of Fourier coefficients tells us how many primes we need as moduli (we leave this analysis to the reader)^{3} . 3

More precisely, the reduction modulo ℓ steps are as follows. The field K can be viewed as Q[x]/(f(x)) where f(x) ∈ Z[x] is the characteristic polynomial of T_{2}. We work only modulo primes such that

1. f(x) has no repeated roots modulo ℓ,

2. ℓ does not divide any denominator involved in our representation of e, and

3. the image of e_{1} in F_{ℓ}[x]/(f(x)) is invertible.

For each such prime, we compute the image \overline{b}_{p} of b_{p} in the reduced Artin ring F_{ℓ}[x]/(f(x)). Then the characteristic polynomial of T_{p} modulo ℓ equals the characteristic polynomial of \overline{b}_{p}. This modular arithmetic is fast and requires negligible storage. Most of the time is spent doing the Chinese Remainder Theorem computations, which we do each time we do a few computations of the characteristic polynomial of T_{p} modulo ℓ. Remark 3.7.4. If k is really large, so that steps 1 and 2 of the algorithm take too long or require too much memory, steps 1 and 2 can be performed modulo the prime ℓ. Since the characteristic polynomial of T_{p,k} modulo ℓ does not depend on any choices, we will still be able to recover the original characteristic polynomial.

3.7.4 How to write down an eigenvector over an extension field

The following algorithm, which was suggested to the author by H. Lenstra, produces an eigenvector defined over an extension of the base field.

Algorithm 3.7.5. Let A be an n×n matrix over an arbitrary field k and suppose that the characteristic polynomial f(x) = x^{n} + ⋯ + a_{1}x + a_{0} of A is irreducible. Let α be a root of f(x) in an algebraic closure \overline{k} of k. Factor f(x) over k(α) as f(x) = (x − α)g(x). Then for any element v ∈ k^{n} the vector g(A)v is either 0 or it is an eigenvector of A with eigenvalue α. The vector g(A)v can be computed by finding Av, A(Av), A(A(Av)), and then using that

g(x) = x^{n−1} + c_{n−2}x^{n−2} + ⋯ + c_{1}x + c_{0},

where the coefficients c_{i} are determined by the recurrence

c_{0} = −\frac{a_{0}}{α} , c_{i} = \frac{c_{i−1} − a_{i}}{α} .

3Say more later.

We prove below that g(A)v ≠ 0 for all vectors v not in a proper subspace of k^{n}. Thus with high probability, a “randomly chosen” v will have the property that g(A)v ≠ 0. Alternatively, if v_{1}, … v_{n} form a basis for k^{n}, then g(A)v_{i} must be nonzero for some i.

Proof. By the Cayley-Hamilton theorem [Lan93, XIV.3] we have that f(A) = 0. Consequently, for any v ∈ k^{n}, we have (A − α)g(A)v = 0 so that Ag(A)v = αv. Since f is irreducible it is the polynomial of least degree satisfied by A and so g(A) ≠ 0. Therefore g(A)v ≠ 0 for all v not in the proper closed subspace ker(g(A)).

3.7.5 Simple example: weight 36, p = 3

We compute the characteristic polynomial of T_{3} acting on S_{36}(1) using the algorithm described above. A basis for M_{36}(1) to precision 6 = 2 dim(S_{36}(1)) is

E_{4}^{9} = 1 + 2160q + 2093040q^{2} + 1198601280q^{3} + 449674832880q^{4}
+ 115759487504160q^{5} + 20820305837344320q^{6} + O(q^{7})
E_{4}^{6}E_{6}^{2} = 1 + 432q − 353808q^{2} − 257501376q^{3} − 19281363984q^{4}
+ 28393576094880q^{5} + 11565037898063424q^{6} + O(q^{7})
E_{4}^{3}E_{6}^{4} = 1 − 1296q + 185328q^{2} + 292977216q^{3} − 52881093648q^{4}
− 31765004621280q^{5} + 1611326503499328q^{6} + O(q^{7})
E_{6}^{6} = 1 − 3024q + 3710448q^{2} − 2309743296q^{3} + 720379829232q^{4}
− 77533149038688q^{5} − 8759475843314112q^{6} + O(q^{7})

The reduced row-echelon form (Miller) basis is:

f_{0} = 1 + 6218175600q^{4} + 15281788354560q^{5} + 9026867482214400q^{6} + O(q^{7})
f_{1} = q + 57093088q^{4} + 37927345230q^{5} + 5681332472832q^{6} + O(q^{7})
f_{2} = q^{2} + 194184q^{4} + 7442432q^{5} − 197264484q^{6} + O(q^{7})
f_{3} = q^{3} − 72q^{4} + 2484q^{5} − 54528q^{6} + O(q^{7})

The matrix of T_{2} with respect to the basis f_{1}, f_{2}, f_{3} is

[T_{2}] = ( \begin{matrix}0 & 34416831456 & 5681332472832 \\ 1 & 194184 & −197264484 \\ 0 & −72 & −54528 \end{matrix})

This matrix has (irreducible) characteristic polynomial

g = x^{3} − 139656x^{2} − 59208339456x − 1467625047588864.

If a is a root of this polynomial, then one finds that

e = (2a + 108984, 2a^{2} + 108984a, a^{2} − 394723152a + 11328248114208)

is an eigenvector with eigenvalue a. The characteristic polynomial of T_{3} is then the characteristic polynomial of e_{3}/e_{1}, which we can compute modulo ℓ for any prime ℓ such that \overline{g} ∈ F_{ℓ}[x] is square free. For example, when ℓ = 11,

\frac{e_{3}}{e_{1}} = \frac{a^{2} + a + 3}{2a^{2} + 7} = 9a^{2} + 2a + 3,

30 3. Modular Forms of Level 1

which has characteristic polynomial

x^{3} + 10x^{2} + 8x + 2.

If we repeat this process for enough primes ℓ and use the Chinese remainder theorem, we find that the characteristic polynomial of T_{3} acting on S_{36}(1) is

x^{3} + 104875308x^{2} − 144593891972573904x − 21175292105104984004394432.

Duality, Rationality, and Integrality

4.1 Modular forms for SL_{2}(Z) and Eisenstein series

Let Γ = Γ_{1}(1) = SL_{2}(Z) and for k ≥ 0 let

M_{k} = {f = ∑_{n=0}^{∞} a_{n}q^{n} : f is a modular form of weight k for Γ}

⊃ S_{k} = {f = ∑_{n=1}^{∞} a_{n}q^{n} : f ∈ M_{k}}

These are finite dimensional C-vector spaces whose dimensions are easily computed. Furthermore, they are generated by familiar elements (see Serre [Ser73] or Lang [Lan95].) The main tool is the formula

∑_{p∈D∪\{∞\}} \frac{1}{e(p)} ord_{p}(f) = \frac{k}{12}

where D is the standard fundamental domain for Γ and

e(p) = \begin{cases}1 & otherwise \\ 2 & if p = i \\ 3 & if p = ρ = e^{2πi/3} \end{cases}

One can alternatively define e(p) as follows. If p = τ and E = C/(Zτ + Z) then e(p) = \frac{1}{2}# Aut(E).

For k ≥ 4 we define the Eisenstein series G_{k} by

G_{k}(q) = \frac{1}{2}ζ(1 − k) + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}.

The map

τ ↦ ∑_{\begin{matrix}(m,n)≠(0,0) \\ m,n∈Z \end{matrix}} \frac{1}{(mτ + n)^{k}}

differs from G_{k} by a constant (see [Ser73, §VII.4.2]). Also, ζ(1 − k) ∈ Q and one

may say, symbolically at least, “ζ(1−k) = ∑_{d=1}^{∞} d^{k−1} = σ_{k−1}(0).” The nth Bernoulli

number B_{n} is defined by the equation

\frac{x}{e^{x} − 1} = ∑_{n=0}^{∞} \frac{B_{n}x^{n}}{n!} .

One can show, using the functional equation for ζ and [Ser73, §VII.4.1], that ζ(1 − k) = −\frac{B_{k}}{k} so the constant coefficient of G_{k} is −\frac{B_{k}}{2k} which is rational.

4.2 Pairings between Hecke algebras and modular forms

In what follows we assume k ≥ 2 to avoid trivialities.. The Hecke operators T_{n} acts on the space M_{k}. Fix a subspace V ⊂ M_{k} which is stable under the action of the T_{n}. Let T = T_{C}(V) = C[…, (T_{n})_{|V} , …] be the C-algebra generated by the endomorphism T_{n} acting on V and note that T(V) is a finite dimensional C-vector space since it is a subspace of End(V) and V is finite dimensional. Recall that T is commutative.

For any Hecke operator T ∈ T and f ∈ V, to make certain arguments notationally more natural, we will often write T (f) = f|T. The ring T is commutative, so it is harmless to use this notation, which suggests both a left and right module structure.

Let T = T_{C}(V), and define the bilinear pairing

V × T → C
⟨f, T ⟩ = a_{1}(f|T)

where f|T = ∑_{n=0}^{∞} a_{n}(f|T)q^{n}. We thus obtain maps

V → Hom(T, C) = T^{∗}
T → Hom(V, C) = V^{∗}.

Theorem 4.2.1. The above maps are isomorphisms.

Proof. It suffices to show that each map is injective. Then since a finite dimensional vector space and its dual have the same dimension the result follows. First suppose f ↦ 0 ∈ Hom(T, C). Then a_{1}(f|T) = 0 for all T ∈ T, so a_{n} = a_{1}(f|T_{n}) = 0 for all n ≥ 1. Thus f is a constant; since k ≥ 2 this implies f = 0.

Next suppose T ↦ 0 ∈ Hom(V, C), then a_{1}(f|T) = 0 for all f ∈ V. Substituting f|T_{n} for f and using the commutativity of T we have

a_{1}((f|T_{n})|T) = 0  for all f, n ≥ 1
a_{1}((f|T)|T_{n}) = 0  by commutativity
a_{n}(f|T) = 0  n ≥ 1
f|T = 0  since k ≥ 2, as above

Thus T = 0 which completes the proof.

Remark 4.2.2. The above isomorphisms are T-equivariant, in the following sense. We endow Hom(T, C) with a T-module structure by letting T ∈ T act on φ ∈ Hom(T, C) by (T · φ)(T′) = φ(T T′). If α : V → Hom(T, C) is the above isomorphism (so α : f ↦ φ_{f} := (T′ ↦ a_{1}(f|T′))), then equivariance is the statement that α(T f) = T α(f). This follows since

α(T f)(T′) = φ_{T f}(T′) = a_{1}(T f|T′) = a_{1}(f|T′T)
= φ_{f}(T′T) = T φ_{f}(T′) = T α(f)(T′).

Similar remarks hold for T → V^{∗}.

4.3 Eigenforms

We continue to assume that k ≥ 2.

Definition 4.3.1. A modular form f ∈ M_{k} is an eigenform for T if f|T_{n} = λ_{n}f for all n ≥ 1 and some complex numbers λ_{n}.

Suppose f is an eigenform, so a_{n}(f) = a_{1}(f|T_{n}) = λ_{n}a_{1}(f). Thus if a_{1}(f) = 0, then a_{n}(f) = 0 for all n ≥ 1, so since k ≥ 2 this implies that f = 0. Thus a_{1}(f) ≠ 0, and we may as well divide through by a_{1}(f) to obtain the normalized eigenform \frac{1}{a_{1}(f)}f. We thus assume that a_{1}(f) = 1, then the formula becomes a_{n}(f) = λ_{n}, so f|T_{n} = a_{n}(f)f, for all n ≥ 1.

Theorem 4.3.2. Let f ∈ V and let ψ be the image of f in Hom(T, C), so ψ(T) = a_{1}(f|T). Then f is a normalized eigenform if and only if ψ is a ring homomorphism.

Proof. First suppose f is a normalized eigenform, so f|T_{n} = a_{n}(f)f. Then

ψ(T_{n}T_{m}) = a_{1}(f|T_{n}T_{m}) = a_{m}(f|T_{n})
= a_{m}(a_{n}(f)f) = a_{m}(f)a_{n}(f)
= ψ(T_{n})ψ(T_{m}),

so ψ is a homomorphism.

Conversely, assume ψ is a homomorphism. Then f|T_{n} = ∑ a_{m}(f|T_{n})q^{m}, so to show that f|T_{n} = a_{n}(f)f we must show that a_{m}(f|T_{n}) = a_{n}(f)a_{m}(f). Recall that ψ(T_{n}) = a_{1}(f|T_{n}) = a_{n}, so

a_{n}(f)a_{m}(f) = a_{1}(f|T_{n})a_{1}(f|T_{m}) = ψ(T_{n})ψ(T_{m})
= ψ(T_{n}T_{m}) = a_{1}(f|T_{n}|T_{m})
= a_{m}(f|T_{n})

as desired.

4.4 Integrality

In the previous sections, we looked at subspaces V ⊂ M_{k} ⊂ C[[q]], with k ≥ 2, and considered the space T = T(V) = C[…, T_{n}, …] ⊂ End_{C} V of Hecke operators on V. We defined a pairing T × V → C by (T, f) ↦ a_{1}(f|T) and showed this pairing is nondegenerate and that it induces isomorphisms T ≅ Hom(V, C) and V ≅ Hom(T, C).

Fix k ≥ 4 and let S = S_{k} be the space of weight k cusp forms with respect to the action of SL_{2}(Z). Let

S(Q) = S_{k} ∩ Q[[q]]
S(Z) = S_{k} ∩ Z[[q]].

Theorem 4.4.1. There is a C-basis of M_{k} consisting of forms with integer coefficients.

Proof. We see this by exhibiting a basis. Recall that for all k ≥ 4

G_{k} = −\frac{B_{k}}{2k} + ∑_{k=1}^{∞} ∑_{d|k} d^{k−1}q^{n}

is the kth Eisenstein series, which is a modular form of weight k, and

E_{k} = −\frac{2k}{B_{k}} · G_{k} = 1 + ⋯

is the normalization that starts with 1. Since the Bernoulli numbers B_{2}, …, B_{8} have 1 as numerator (this isn’t always the case, B_{10} = \frac{5}{66}) we see that E_{4} and E_{6} have coefficients in Z and constant term 1. Furthermore one shows (see [Ser73, §VII.3.2]) by dimension and independence arguments that the modular forms

{E_{4}^{a}E_{6}^{b} : 4a + 6b = k}

form a basis for M_{k}.

4.5 A Result from Victor Miller’s thesis

Set d = dim_{C} S_{k}. Victor Miller showed in his thesis (see [Lan95], Ch. X, Theorem 4.4) that there exists

f_{1}, …, f_{d} ∈ S_{k}(Z) such that a_{i}(f_{j}) = δ_{ij}

for 1 ≤ i, j ≤ d. The f_{i} then form a basis for S_{k}(Z). Example 4.5.1. The space S_{36}(Z) has basis

f_{1} = q + 57093088q^{4} + 37927345230q^{5} + 5681332472832q^{6} + ⋯
f_{2} = q^{2} + 194184q^{4} + 7442432q^{5} − 197264484q^{6} + 722386944q^{7} ⋯
f_{3} = q^{3} − 72q^{4} + 2484q^{5} − 54528q^{6} + 852426q^{7} − 10055232q^{8} + ⋯

Let T = Z[…, T_{n}, …] ⊂ End(S_{k}) be the Hecke algebra associated to S_{k}. Miller’s thesis implies the following result about T.

Proposition 4.5.2. We have T = ⨁_{i=1}^{d} ZT_{i}, as Z-modules.

Proof. To see that T_{1}, …, T_{d} ∈ T = T(S_{k}) are linearly independent over C, suppose ∑_{i=1}^{d} c_{i}T_{i} = 0. Then

0 = a_{1} (f_{j}| ∑ c_{i}T_{i}) = ∑_{i} c_{i}a_{i}(f_{j}) = ∑_{i} c_{i}δ_{ij} = c_{j}.

From the isomorphism T ≅ Hom(S_{k}, C) we know that dim_{C} T = d, so we can write any T_{n} as a C-linear combination

T_{n} = ∑_{i=1}^{d} c_{n_{i}}T_{i}, c_{n_{i}} ∈ C.

But

Z ∋ a_{n}(f_{j}) = a_{1}(f_{j}|T_{n}) = ∑_{i=1}^{d} c_{n_{i}}a_{1}(f_{j}|T_{i}) = ∑_{i=1}^{d} c_{n_{i}}a_{i}(f_{j}) = c_{n_{j}}

so the c_{n_{i}} all lie in Z, which completes the proof.

Thus T is an integral Hecke algebra of finite rank d over Z. We have a map

S(Z) × T → Z
(f, T) ↦ a_{1}(f|T)

which induces an embedding

S(Z) ↪ Hom(T, Z) ≈ Z^{d}.

Remark 4.5.3. The map S(Z) ↪ Hom(T, Z) is an isomorphism of T-modules, since if φ ∈ Hom(T, Z), then f = ∑_{j=1}^{d} φ(T_{j})f_{j} ∈ S(Z) maps to φ.

4.6 The Petersson inner product

The main theorem is

Theorem 4.6.1. The T_{n} ∈ T(S_{k}) are all diagonalizable over C.

To prove this, we note that S_{k} supports a nondegenerate positive definite Hermitean inner product (the Petersson inner product)

(f, g) ↦ ⟨f, g⟩ ∈ C

such that

⟨f|T_{n}, g⟩ = ⟨f, g|T_{n}⟩.  (4.6.1)

We need some background facts.

Definition 4.6.2. An operator T is normal if it commutes with its adjoint, thus T T^{∗} = T^{∗}T.

It follows from (4.6.1) that T_{n}^{∗} = T_{n}, so T_{n} is clearly normal.

Theorem 4.6.3. A normal operator is diagonalizable.

Thus each T_{n} is diagonalizable.

Theorem 4.6.4. A commuting family of semisimple (=diagonalizable) operators can be simultaneously diagonalized.

Since the T_{n} commute this implies S_{k} has a basis consisting of normalized eigenforms f. Their eigenvalues are real since

a_{n}(f)⟨f, f⟩ = ⟨a_{n}(f)f, f⟩ = ⟨f|T_{n}, f⟩  (4.6.2)
= ⟨f, a_{n}(f)f⟩ = \overline{a_{n}(f)}⟨f, f⟩.  (4.6.3)

Proposition 4.6.5. The coefficients a_{n} of the eigenforms are totally real algebraic integers.

Proof. Theorem 4.4.1 implies that there is a basis for S_{k} with integer coefficients. With respect to a basis for S_{k}(Z) = S_{k} ∩ Z[[q]], the matrices of the Hecke opeators T_{n} all have integer entries, so their characteristic polynomials are monic polynomials with integer coefficients. The roots of these polynomials are all real (by (4.6.2)), so the roots are totally real algebraic integers.

Let

𝔥 = {x + iy : x, y ∈ R, and y > 0}

be the upper half plane.

If α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(R) then ( \begin{matrix}a & b \\ c & d \end{matrix} ) acts on 𝔥 by

(\begin{matrix}a & b \\ c & d \end{matrix}) : z ↦ \frac{az + b}{cz + d}.

Lemma 4.6.6.

Im (\frac{az + b}{cz + d}) = \frac{det(α)}{|cz + d|^{2}} y.

Proof. Apply the identity Im(w) = \frac{1}{2i}(w−\overline{w}) to the left hand side and simplify.

Proposition 4.6.7. The volume form \frac{dx∧dy}{y^{2}} is invariant under the action of

GL_{2}^{+}(R) = {α ∈ GL_{2}(R) : det(α) > 0}.

Proof. Differentiating α = \frac{az+b}{cz+d} gives

d (\frac{az + b}{cz + d}) = \frac{a(cz + d)dz − c(az + b)dz}{(cz + d)^{2}}

= \frac{(ad − bc)dz}{(cz + d)^{2}}

= \frac{det(α)}{(cz + d)^{2}} dz

Thus, under the action of α, dz ∧ d\overline{z} takes on a factor of

\frac{det(α)^{2}}{(cz + d)^{2}(c\overline{z} + d)^{2}} = (\frac{det(α)}{|cz + d|^{2}} )^{2}.

Definition 4.6.8. The Petersson inner product of forms f, g ∈ S_{k} is defined by

⟨f, g⟩ = ∫_{Γ\𝔥}(f(z)\overline{g(z)}y^{k})\frac{dx ∧ dy}{y^{2}} ,

where Γ = SL_{2}(Z).

Integrating over Γ\𝔥 can be taken to mean integrating over a fundamental domain for the action of 𝔥. Showing that the operators T_{n} are self-adjoint with respect to the Petersson inner product is a harder computation than one might think (see [Lan95, §III.4]).

38 4. Duality, Rationality, and Integrality

Analytic Theory of Modular Curves

5.1 The Modular group

This section very closely follows Sections 1.1–1.2 of [Ser73]. We introduce the modular group G = PSL_{2}(Z), describe a fundamental domain for the action of G on the upper half plane, and use it to prove that G is generated by

S = (\begin{matrix}0 & − 1 \\ 1 & 0 \end{matrix}) and T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) .

5.1.1 The Upper half plane

Let

𝔥 = {z ∈ C : Im(z) > 0}

be the open complex upper half plane. The group

SL_{2}(R) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) : a, b, c, d ∈ R and ad − bc = 1\}

acts by linear fractional transformations (z ↦ (az + b)/(cz + d)) on C ∪ {∞}. Suppose g ∈ SL_{2}(R) and z ∈ 𝔥. Then Lemma 4.6.6 implies that Im(gz) = \frac{Im(z)}{|cz+d|^{2}} , so SL_{2}(R) acts on 𝔥.

The only element of SL_{2}(R) that acts trivially on 𝔥 is −1, so

G = PSL_{2}(Z) = SL_{2}(Z)/⟨−1⟩

FIGURE 5.1.1. The upper half plane 𝔥

acts faithfully on 𝔥. Let S and T be as above and note that S and T induce the linear fractional transformations z ↦ −1/z and z ↦ z + 1, respectively. In fact, S and T generate G.

5.2 Points on modular curves parameterize elliptic curves with extra structure

The classical theory of the Weierstass ℘-function sets up a bijection between isomorphism classes of elliptic curves over C and isomorphism classes of onedimensional complex tori C/Λ. Here Λ is a lattice in C, i.e., a free abelian group Zω_{1} + Zω_{2} of rank 2 such that Rω_{1} + Rω_{2} = C.

Any homomorphism φ of complex tori C/Λ_{1} → C/Λ_{2} is determined by a Clinear map T : C → C that sends Λ_{1} into Λ_{2}.

Lemma 5.2.1. Suppose φ : C/Λ_{1} → C/Λ_{2} is nonzero. Then the kernel of φ is isomorphic to Λ_{2}/T (Λ_{1}).

Lemma 5.2.2. Two complex tori C/Λ_{1} and C/Λ_{2} are isomorphic if and only if there is a complex number α such that αΛ_{1} = Λ_{2}.

Proof. Any C-linear map C → C is multiplication by a scalar α ∈ C.

Suppose Λ = Zω_{1} + Zω_{2} is a lattice in C, and let τ = ω_{1}/ω_{2}. Then Λ_{τ} = Zτ + Z defines an elliptic curve that is isomorphic to the elliptic curve determined by Λ. By replacing ω_{1} by −ω_{1}, if necessary, we may assume that τ ∈ 𝔥. Thus every elliptic curve is of the form E_{τ} = C/Λ_{τ} for some τ ∈ 𝔥 and each τ ∈ 𝔥 determines an elliptic curve.

Proposition 5.2.3. Suppose τ, τ′ ∈ 𝔥. Then E_{τ} ≈ E_{τ′} if and only if there exists g ∈ SL_{2}(Z) such that τ = g(τ′). Thus the set of isomorphism classes of elliptic curves over C is in natural bijection with the orbit space SL_{2}(Z)\𝔥.

Proof. Suppose E_{τ} ≈ E_{τ′}. Then there exists α ∈ C such that αΛ_{τ} = Λ_{τ′}, so ατ = aτ′ + b and α1 = cτ′ + d for some a, b, c, d ∈ Z. The matrix g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) has determinant ±1 since aτ′ +b and cτ′ +d form a basis for Zτ′ +Z; this determinant is positive because g(τ′) = τ and τ, τ′ ∈ 𝔥. Thus det(g) = 1, so g ∈ SL_{2}(Z).

Conversely, suppose τ, τ′ ∈ 𝔥 and g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) is such that

τ = g(τ′) = \frac{aτ′ + b}{cτ′ + d}.

Let α = cτ′ + d, so ατ = aτ′ + b. Since det(g) = 1, the scalar α defines an isomorphism from Λ_{τ} to Λ_{τ′}, so E_{τ} ≈ E′_{τ}, as claimed.

Let E = C/Λ be an elliptic curve over C and N a positive integer. Using Lemma 5.2.1, we see that

E[N] := \{x ∈ E : Nx = 0\} ≅ (\frac{1}{N} Λ) /Λ ≅ (Z/NZ)^{2}.

If Λ = Λ_{τ} = Zτ + Z, this means that τ/N and 1/N are a basis for E[N].

Suppose τ ∈ 𝔥 and recall that E_{τ} = C/Λ_{τ} = C/(Zτ + Z). To τ, we associate three “level N structures”. First, let C_{τ} be the subgroup of E_{τ} generated by 1/N. Second, let P_{τ} be the point of order N in E_{τ} defined by 1/N ∈ \frac{1}{N} Λ_{τ}. Third, let Q_{τ} be the point of order N in E_{τ} defined by τ/N, and consider the basis (P_{τ} , Q_{τ}) for E[N].

In order to describe the third level structure, we introduce the Weil pairing

e : E[N] × E[N] → Z/NZ

as follows. If E = C/(Zω_{1} + Zω_{2}) with τ = ω_{1}/ω_{2} ∈ 𝔥, and P = aω_{1}/N + bω_{2}/N, Q = cω_{1}/N + dω_{2}/N, then

e(P, Q) = ad − bc ∈ Z/NZ.

Note that e does not depend on choice of basis ω_{1}, ω_{2} for Λ. Also if C/Λ ≅ C/Λ′ via multiplication by α, and P, Q ∈ (C/Λ)[N], then we have e(α(P), α(Q)) = e(P, Q), so e does not depend on the choice of Λ. In particular, P_{τ} and Q_{τ} map to P, Q via the map E_{τ} → E given by multiplication by ω_{2}, so e(P_{τ} , Q_{τ}) = −1 ∈ Z/NZ. Remark 5.2.4. There is a canonical Nth root of 1 in C, namely ζ = e^{2πi/N}. Using ζ as a canonical generator of μ_{N}, we can view the transcendental Weil pairing indifferently as a map with values in Z/NZ or as a map with values in μ_{N}. However, for generalizations it is important to use μ_{N} rather than Z/NZ. There are several intrinsic algebraic definitions of the Weil pairing on N-division points for an elliptic curve (or, more generally, an abelian variety) over a field k whose characteristic is prime to N. In all cases, the Weil pairing takes values in the group of Nth roots of unity with values in the algebraic closure of k. The various definitions all coincide “up to sign” in the sense that any two of them either coincide or are inverse to each other. There are discussions of the Weil pairing in [Kat81, §5.2] and [Sil92, III.8]. The Weil pairing is bilinear, alternating, non-degenerate, Galois invariant, and maps surjectively onto μ_{N}.

We next consider the three modular curves X_{0}(N), X_{1}(N), and X(N), associated to the congruence subgroups Γ_{0}(N), Γ_{1}(N) and Γ(N). Recall that Γ_{0}(N) is the subgroup of ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) congruent to ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix}) modulo N, that Γ_{1}(N) consists of matrices congruent to ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) modulo N, and Γ(N) consists of matrices congruent to ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) modulo N.

Theorem 5.2.5. Let N be a positive integer.

1. The non-cuspidal points on X_{0}(N) correspond to isomorphism classes of pairs (E, C) where C is a cyclic subgroup of E of order N. (Two pairs (E, C), (E′, C′) are isomorphic if there is an isomorphism φ : E → E′ such that φ(C) = C′.)

2. The non-cuspidal points on X_{1}(N) correspond to isomorphism classes of pairs (E, P) where P is a point on E of exact order N. (Two pairs (E, P) and (E′, P′) are isomorphic if there is an isomorphism φ : E → E′ such that φ(P) = P′.)

3. The non-cuspidal points on X(N) correspond to isomorphism classes of triples (E, P, Q) where P, Q is a basis for E[N] such that e(P, Q) = −1 ∈ Z/NZ. (Triples (E, P, Q) and (E, P′, Q′) are isomorphic if there is an isomorphism φ : E → E′ such that φ(P) = P′ and φ(Q) = Q′.)

This theorem follows from Propositions 5.2.8 and 5.2.9 below, whose proofs make extensive use of the following lemma.

Lemma 5.2.6. The reduction map SL_{2}(Z) → SL_{2}(Z/NZ) is surjective.

Proof. By considering the bottom two entries of an element of SL_{2}(Z/NZ) and using the extended Euclidean algorithm, it suffices to prove that if c, d ∈ Z and gcd(c, d, N) = 1, then there exists α ∈ Z such that gcd(c, d + αN) = 1. This suffices because of g_{1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and g_{2} = ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) are both in SL_{2}(Z/NZ), then a simple calculation shows that g_{1}g_{2}^{−1} = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ), and every element of SL_{2}(Z/NZ) of the form ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) is in the image of the reduction map.

Let c_{0} = ∏_{ℓ|gcd(c,d)} ℓ^{ord_{ℓ}(c)} and c_{1} = c/c_{0}. We have constructed c_{0} and c_{1} so that gcd(d, c_{1}) = 1 and gcd(c_{0}, c_{1}) = 1; also, gcd(c_{0}, N) = 1 since if ℓ | gcd(c_{0}, N), then ℓ | gcd(c, d, N) = 1. Because gcd(Nc_{1}, c_{0}) = 1, the class of Nc_{1} generates the additive group Z/c_{0}Z, so there exists m such that m · Nc_{1} ≡ 1 − d (mod c_{0}). Thus d + mc_{1}N ≡ 1 (mod c_{0}) and d + mc_{1}N ≡ d (mod c_{1}), so d + mc_{1}N is a unit modulo both c_{0} and c_{1}, hence gcd(d + mc_{1}N, c) = 1, which proves the lemma.

Remark 5.2.7. The analogue of Lemma 5.2.6 is also true for SL_{m}(Z) → SL_{m}(Z/NZ) for any integer m. See [Shi94, ?].

Proposition 5.2.8. Let E be an elliptic curve over C. If C is a cyclic subgroup of E of order N, then there exists τ ∈ 𝔥 such that (E, C) is isomorphic to (E_{τ} , C_{τ}). If P is a point on E of order N, then there exists τ ∈ C such that (E, P) is isomorphic to (E_{τ} , P_{τ}). If P, Q is a basis for E[N] and e(P, Q) = −1 ∈ Z/NZ, then there exists τ ∈ C such that (E, P, Q) is isomorphic to (E_{τ} , P_{τ} , Q_{τ}).

Proof. Write E = C/Λ with Λ = Zω_{1} + Zω_{2} and ω_{1}/ω_{2} ∈ 𝔥.

Suppose P = aω_{1}/N + bω_{2}/N is a point of order N. Then gcd(a, b, N) = 1, since otherwise P would have order strictly less than N, a contradiction. As in Lemma 5.2.6, we can modify a and b by adding multiples of N to them, so that P = aω_{1}/N +bω_{2}/N and gcd(a, b) = 1. There exists c, d ∈ Z such that ad−bc = 1, so ω′_{1} = aω_{1} + bω_{2} and ω′_{2} = cω_{1} + dω_{2} form a basis for Λ, and C is generated by P = ω′_{1}/N. If necessary, replace ω′_{2} by −ω′_{2} so that τ = ω′_{2}/ω′_{1} ∈ 𝔥. Then (E, P) is isomorphic to (E_{τ} , P_{τ}). Also, if C is the subgroup generated by P, then (E, C) is isomorphic to (E_{τ} , C_{τ}).

Suppose P = aω_{1}/N + bω_{2}/N and Q = cω_{1}/N + dω_{2}/N are a basis for E[N] with e(P, Q) = −1. Then the matrix ( \begin{matrix}a & b \\ −c & −d \end{matrix} ) has determinant 1 modulo N, so because the map SL_{2}(Z) → SL_{2}(Z/NZ) is surjective, we can replace a, b, c, d by integers that are equivalent to them modulo N (so P and Q are unchanged) so that ad − bc = −1. Thus ω′_{1} = aω_{1} + bω_{2} and ω′_{2} = cω_{1} + dω_{2} form a basis for Λ. Let

τ = ω′_{2}/ω′_{1} = \frac{c\frac{ω_{1}}{ω_{2}} + d}{a\frac{ω_{1}}{ω_{2}} + b}.

Then τ ∈ 𝔥 since ω_{1}/ω_{2} ∈ 𝔥 and ( \begin{matrix}c & d \\ a & b \end{matrix} ) has determinant +1. Finally, division by ω′_{1} defines an isomorphism E → E_{τ} that sends P to 1/N and Q to τ/N.

The following proposition completes the proof of Theorem 5.2.5.

5.3 The Genus of X(N) 43

Proposition 5.2.9. Suppose τ, τ′ ∈ 𝔥. Then (E_{τ} , C_{τ}) is isomorphic (E_{τ′}, C_{τ′}) if and only if there exists g ∈ Γ_{0}(N) such that g(τ) = τ′. Also, (E_{τ} , P_{τ}) is isomorphic (E_{τ′}, P_{τ′}) if and only if there exists g ∈ Γ_{1}(N) such that g(τ) = τ′. Finally, (E_{τ} , P_{τ} , Q_{τ}) is isomorphic (E_{τ′}, P_{τ′}, Q_{τ′}) if and only if there exists g ∈ Γ(N) such that g(τ) = τ′.

Proof. We prove only the first assertion, since the others are proved in a similar way. Suppose (E_{τ} , C_{τ}) is isomorphic to (E′_{τ} , C′_{τ}). Then there is λ ∈ C such that λΛ_{τ} = Λ_{τ′}, and multiplication by λ sends C_{τ} onto C′_{τ}. Thus λτ = aτ′ + b and λ1 = cτ′ +d with g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) (as we saw in the proof of Proposition 5.2.3). Dividing the second equation by N we get λ\frac{1}{N} = \frac{c}{N} τ′ + \frac{d}{N}, which lies in Λ_{τ′} = Zτ′ + \frac{1}{N} Z, by hypothesis. Thus c ≡ 0 (mod N), so g ∈ Γ_{0}(N), as claimed. For the converse, note that if N | c, then \frac{c}{N} τ′ + \frac{d}{N} ∈ Λ_{τ′}.

5.3 The Genus of X(N)

Let N be a positive integer. The aim of this section is to establish some facts about modular curves associated to congruence subgroups and compute the genus of X(N). Similar methods can be used to compute the genus of X_{0}(N) and X_{1}(N) (see Chapter 10).

The groups Γ_{0}(1), Γ_{1}(1), and Γ(1) are all equal to SL_{2}(Z), so X_{0}(1) = X_{1}(1) = X(1) = P^{1}. Since P^{1} has genus 0, we know the genus for each of these three cases. For general N we obtain the genus by determining the ramification of the corresponding cover of P^{1} and applying the Hurwitz formula, which we assume the reader is familiar with, but which we now recall.

Suppose f : X → Y is a surjective morphism of Riemann surfaces of degree d. For each point x ∈ X, let e_{x} be the ramification exponent at x, so e_{x} = 1 precisely when f is unramified at x, which is the case for all but finitely many x. (There is a point over y ∈ Y that is ramified if and only if the cardinality of f^{−1}(y) is less than the degree of f.) Let g(X) and g(Y) denote the genera of X and Y, respectively.

Theorem 5.3.1 (Hurwitz Formula). Let f : X → Y be as above. Then

2g(X) − 2 = d(2g(Y) − 2) + ∑_{x∈X}(e_{x} − 1).

If X → Y is Galois, so the e_{x} in the fiber over each fixed y ∈ Y are all equal to a fixed value e_{y}, then this formula becomes

2g(X) − 2 = d (2g(Y) − 2 + ∑_{y∈Y} (1 − \frac{1}{e_{y}} )) .

Let X be one of the modular curves X_{0}(N), X_{1}(N), or X(N) corresponding to a congruence subgroup Γ, and let Y = X(1) = P^{1}. There is a natural map f : X → Y got by sending the equivalence class of τ modulo the congruence subgroup Γ to the equivalence class of τ modulo SL_{2}(Z). This is “the” map X → P^{1} that we mean everywhere below.

Because PSL_{2}(Z) acts faithfully on 𝔥, the degree of f is the index in PSL_{2}(Z) of the image of Γ in PSL_{2}(Z). Using Lemma 5.2.6 that the map SL_{2}(Z) → SL_{2}(Z/NZ) is surjective, we compute these indices, and obtain the following:

Proposition 5.3.2. Suppose N > 2. The degree of the map X_{0}(N) → P^{1} is N ∏_{p|N}(1 + 1/p). The degree of the map X_{1}(N) → P^{1} is \frac{1}{2}N^{2} ∏_{p|N}(1 − 1/p^{2}). The degree of the map from X(N) → P^{1} is \frac{1}{2}N^{3} ∏_{p|N}(1 − 1/p^{2}). If N = 2, then the degrees are 3, 3, and 6, respectively.

Proof. This follows from the discussion above, and the observation that for N > 2 the groups Γ(N) and Γ_{1}(N) do not contain −1 and the group Γ_{0}(N) does.

Proposition 5.3.3. Let X be X_{0}(N), X_{1}(N) or X(N). Then the map X → P^{1} is ramified at most over ∞ and the two points corresponding to elliptic curves with extra automorphisms (i.e., the two elliptic curves with j-invariants 0 and 1728).

Proof. Since we have a tower X(N) → X_{1}(N) → X_{0}(N) → P^{1}, it suffices to prove the assertion for X = X(N). Since we do not claim that there is no ramification over ∞, we may restrict to Y (N). By Theorem 5.2.5, the points on Y (N) correspond to isomorphism classes of triples (E, P, Q), where E is an elliptic curve over C and P, Q are a basis for E[N]. The map from Y (N) to P^{1} sends the isomorphism class of (E, P, Q) to the isomorphism class of E. The equivalence class of (E, P, Q) also contains (E, −P, −Q), since −1 : E → E is an isomorphism. The only way the fiber over E can have cardinality smaller than the degree is if there is an extra equivalence (E, P, Q) → (E, φ(P), φ(Q)) with φ an automorphism of E not equal to ±1. The theory of CM elliptic curves shows that there are only two isomorphism classes of elliptic curves E with automorphisms other than ±1, and these are the ones with j-invariant 0 and 1728. This proves the proposition.

Theorem 5.3.4. For N > 2, the genus of X(N) is

g(X(N)) = 1 + \frac{N^{2}(N − 6)}{24} ∏_{p|N} (1 − \frac{1}{p^{2}} ) ,

where p runs through the prime divisors of N. For N = 1, 2, the genus is 0.

Thus if g_{N} = g(X(N)), then g_{1} = g_{2} = g_{3} = g_{4} = g_{5} = 0, g_{6} = 1, g_{7} = 3, g_{8} = 5, g_{9} = 10, g_{389} = 2414816, and g_{2003} = 333832500.

Proof. Since Γ(N) is a normal subgroup of SL_{2}(Z), it follows that X(N) is a Galois covering of X(1) = P^{1}, so the ramification indices e_{x} are all the same for x over a fixed point y ∈ P^{1}; we denote this common index by e_{y}. The fiber over the curve with j-invariant 0 has size one-third of the degree, since the automorphism group of the elliptic curve with j-invariant 0 has order 6, so the group of automorphisms modulo ±1 has order three, hence e_{0} = 3. Similarly, the fiber over the curve with j-invariant 1728 has size half the degree, since the automorphism group of the elliptic curve with j-invariant 1728 is cyclic of order 4, so e_{1728} = 2.

To compute the ramification degree e_{∞} we use the orbit stabilizer theorem. The fiber of X(N) → X(1) over ∞ is exactly the set of Γ(N) equivalence classes of cusps, which is Γ(N)∞, Γ(N)g_{2}∞, …, Γ(N)g_{r}∞, where g_{1} = 1, g_{2}, …, g_{r} are coset representatives for Γ(N) in SL_{2}(Z). By the orbit-stabilizer theorem, the number of cusps equals #(Γ(1)/Γ(N))/#S, where S is the stabilizer of Γ(N)∞ in Γ(1)/Γ(N) ≅ SL_{2}(Z/NZ). Thus S is the subgroup \{± ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) : 0 ≤ n < N − 1\}, which has order 2N. Since the degree of X(N) → X(1) equals #(Γ(1)/Γ(N))/2, the number of cusps is the degree divided by N. Thus e_{∞} = N.

5.3 The Genus of X(N) 45

The Hurwitz formula for X(N) → X(1) with e_{0} = 3, e_{1728} = 2, and e_{∞} = N, is

2g(X(N)) − 2 = d (0 − 2 + (1 − \frac{1}{3} + 1 − \frac{1}{2} + 1 − \frac{1}{N} )) ,

where d is the degree of X(N) → X(1). Solving for g(X(N)) we obtain

2g(X(N)) − 2 = d (1 − \frac{5}{6} − \frac{1}{N} ) = d (\frac{N − 6}{6N}) ,

so

g(X(N)) = 1 + \frac{d}{2} (\frac{N − 6}{6N}) = \frac{d}{12N} (N − 6) + 1.

Substituting the formula for d from Proposition 5.3.2 yields the claimed formula.

46 5. Analytic Theory of Modular Curves

Modular Curves

6.1 Cusp Forms

Recall that if N is a positive integer we define the congruence subgroups Γ(N) ⊂ Γ_{1}(N) ⊂ Γ_{0}(N) by

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : c ≡ 0 (mod N)\}
Γ_{1}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : a ≡ d ≡ 1, c ≡ 0 (mod N)\}
Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) (mod N)\}.

Let Γ be one of the above subgroups. One can give a construction of the space S_{k}(Γ) of cusp forms of weight k for the action of Γ using the language of algebraic geometry. Let X_{Γ} = Γ\H^{∗} be the upper half plane (union the cusps) modulo the action of Γ. Then X_{Γ} can be given the structure of compact Riemann surface and S_{2}(Γ) = H^{0}(X_{Γ}, Ω^{1}) where Ω^{1} is the sheaf of differential 1-forms on X_{Γ}. This is true since an element of H^{0}(X_{Γ}, Ω^{1}) is a differential form f(z)dz, holomorphic on H and the cusps, which is invariant with respect to the action of Γ. If γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ then

d(γ(z))/dz = (cz + d)^{−2}

so

f(γ(z))d(γ(z)) = f(z)dz

if and only if f satisfies the modular condition

f(γ(z)) = (cz + d)^{2}f(z).

6.2 Modular curves

Recall from Section 5.2 that SL_{2}(Z)\𝔥 parameterizes isomorphism classes of elliptic curves, and the other congruence subgroups also give rise to similar parameterizations. Thus Γ_{0}(N)\𝔥 parameterizes isomorphism classes of pairs (E, C) where E is an elliptic curve and C is a cyclic subgroup of order N, and Γ_{1}(N)\H parameterizes isomorphism classes of pairs (E, P) where E is an elliptic curve and P is a point of exact order N.

We can specify a point of exact order N on an elliptic curve E by giving an injection Z/NZ ↪ E[N], or equivalently, an injection μ_{N} ↪ E[N] where μ_{N} denotes the Nth roots of unity. Then Γ(N)\𝔥 parameterizes isomorphism classes of pairs (E, {α, β}), where {α, β} is a basis for E[N] ≈ (Z/NZ)^{2}.

The above quotients spaces are called moduli spaces for the moduli problem of determining equivalence classes of pairs (E+ extra structure).

6.3 Classifying Γ(N)-structures

Definition 6.3.1. Let S be an arbitrary scheme. An elliptic curve E/S is a proper smooth curve

\xymatrix{E \ar[d]_{f} \\ S}

with geometrically connected fibers all of genus one, give with a section “0”.

Loosely speaking, proper is a generalization of projective and smooth generalizes nonsingularity. See Hartshorne [Har77, III.10] for the precise definitions.

Definition 6.3.2. Let S be any scheme and E/S an elliptic curve. A Γ(N)- structure on E/S is a group homomorphism

φ : (Z/NZ)^{2} → E[N](S)

whose image “generates” E[N](S). (A good reference is [KM85, Ch. 3].)

Define a functor from the category of Q-schemes to the category of sets by sending a scheme S to the set of isomorphism classes of pairs

(E, Γ(N)-structure)

where E is an elliptic curve defined over S and isomorphisms (preserving the Γ(N)-structure) are taken over S. An isomorphism preserves the Γ(N)-structure if it takes the two distinguished generators to the two distinguished generators in the image (in the correct order).

Theorem 6.3.3. For N ≥ 4 the functor defined above is representable and the object representing it is the modular curve X corresponding to Γ(N).

What this means is that given a Q-scheme S, the set

X(S) = Mor_{Q-schemes}(S, X)

is isomorphic to the image of the functor’s value on S.

There is a natural way to map a pair (E, Γ(N)-structure) to an Nth root of unity. Recall from Section 5.2 that if P, Q are the distinguished basis of E[N] we send the pair (E, Γ(N)-structure) to

e_{N}(P, Q) ∈ μ_{N}

where e_{N} : E[N] × E[N] → μ_{N} is the Weil pairing.

6.4 More on integral Hecke operators

Consider the algebra of integral Hecke operators T = T_{Z} on the space of cusp forms S_{k}(C) with respect to the action of the full modular group SL_{2}(Z). Our goal is to see why T ≅ Z^{d} where d = dim_{C} S_{k}(C).

Suppose A ⊂ C is any subring of C and let

T_{A} = A[…, T_{n}, …] ⊂ End_{C} S_{k}.

We have a natural map

T_{A} ⊗_{A} C → T_{C}

but we do not yet know that it is an isomorphism.

6.5 Complex conjugation

We have a conjugate linear map on functions

f(τ) ↦ \overline{f(−\overline{τ})}.

Since \overline{(e^{−2πi\overline{τ}})} = e^{2πiτ}, it follows that under the above map,

∑_{n=1}^{∞} a_{n}q^{n} ↦ ∑_{n=1}^{∞} \overline{a_{n}}q^{n},

so it is reasonable to call this map “complex conjugation”. Furthermore, if we know that

S_{k}(C) = C ⊗_{Q} S_{k}(Q),

then it follows that complex conjugation takes S_{k}(C) into S_{k}(C). To see this note that if we have the above equality then every element of S_{k}(C) is a C-linear combination of elements of S_{k}(Q); conversely, it is clear that the set of such Clinear combinations is invariant under the action of complex conjugation.

6.6 Isomorphism in the real case

Proposition 6.6.1. T_{R} ⊗_{R} C ≅ T_{C}, as C-vector spaces.

Proof. Since S_{k}(R) = S_{k}(C) ∩ R[[q]] and since Theorem 4.4.1 assures us that there is a C-basis of S_{k}(C) consisting of forms with integral coefficients, we see that S_{k}(R) ≈ R^{d} where d = dim_{C} S_{k}(C). (Any element of S_{k}(R) is a C-linear combination of the integral basis, hence equating real and imaginary parts, an Rlinear combination of the integral basis, and the integral basis stays independent over R.) By considering the explicit formula for the action of the Hecke operators T_{n} on S_{k} (see Section 3.5.2) we see that T_{R} leaves S_{k}(R) invariant, thus

T_{R} = R[…, T_{d}, …] ⊂ End_{R} S_{k}(R).

In Section 4.2 we defined a perfect pairing

T_{C} × S_{k}(C) → C.

By restricting to R we again obtain a perfect pairing, so we see that T_{R} ≈ S_{k}(R) ≈ R^{d} which implies that T_{R} ⊗_{R} C ⟶^{∼} T_{C}.

The above argument also shows that S_{k}(C) ≅ S_{k}(R) ⊗_{R} C, so complex conjugation is defined over R.

6.7 The Eichler-Shimura isomorphism

Our goal in this section is to briefly outline a homological interpretation of S_{k} = S_{k}(C). For details see [Lan95, VI], the original paper [Shi59], or [Shi94, VIII].

How is the space S_{k}(C) of cusp forms related to the cohomology group H^{1}(X_{Γ}, R)? Suppose k = 2 and Γ ⊂ SL_{2}(Z) is a congruence subgroup, and let X_{Γ} = \overline{Γ\H} be the Riemann surface obtained by compactifying the upper half plane modulo the action of Γ. Then S_{k}(C) = H^{0}(X_{Γ}, Ω^{1}) so we have a pairing

H_{1}(X_{Γ}, Z) × S_{k}(C) → C

given by integration

(γ, ω) ↦ ∫_{γ} ω.

This gives an embedding

Z^{2d} ≈ H_{1}(X_{Γ}, Z) ↪ Hom_{C}(S_{k}(C), C) ≅ C^{d}

of a “lattice” in C^{d} (we write “lattice” because we have not shown that the image of Z^{2d} is really a lattice, i.e., has R span equal to C^{d}). Passing to the quotient and compactifying, we obtain a complex torus called the Jacobian of X_{Γ}. Again, using the above pairing, we obtain an embedding

C^{d} ≈ S_{k}(C) ↪ Hom(H_{1}(X_{Γ}, Z), C) ≅ C^{2d}

which, upon taking the real part, gives

S_{k}(C) → Hom(H_{1}(X_{Γ}, Z), R) ≅ H^{1}(X_{Γ}, R) ≅ H_{p}^{1}(Γ, R)

where H_{p}^{1}(Γ, R) denotes the parabolic group cohomology of Γ with respect to the trivial action. It is this result, that we may view S_{k}(C) as the cohomology group H_{p}^{1}(Γ, R), which was alluded to above.

Shimura generalized this for arbitrary k ≥ 2, and showed that

S_{k}(C) ≅ H_{p}^{1}(Γ, V_{k}),

where V_{k} is a k − 1 dimensional R-vector space. The isomorphism is (approximately) the following: f ∈ S_{k}(C) is sent to the cocycle map

γ ↦ Re ∫_{τ_{0}}^{γτ_{0}} f(τ)τ^{i}dτ, i = 0, …, k − 2.

Let W = R ⊕ R, then Γ acts on W by

(\begin{matrix}a & b \\ c & d \end{matrix}) : (\begin{matrix}x \\ y \end{matrix}) ↦ (\begin{matrix}ax + by \\ cx + dy \end{matrix})

so Γ acts on

V_{k} = Sym^{k−2} W = W^{⊗k−2}/S_{k−2}

where S_{k−2} is the symmetric group on k − 2 symbols (note that dim V_{k} = k − 1). Let

L = H_{p}^{1}(Γ, Sym^{k−2}(Z ⊕ Z)).

Under the isomorphism

S_{k}(C) ≅ H_{p}^{1}(Γ, R),

L is a sublattice of S_{k}(C) of Z-rank 2d = 2 dim_{C} S_{k}(C) which is T_{n}-stable for all n. Thus we have an embedding

T_{Z} = T ↪ End L,

so T_{R} ⊂ End_{R}(L ⊗ R) and T_{Z} ⊗_{Z} R ≅ T_{R}, which has rank d.

52 6. Modular Curves

Modular Symbols

This chapter is about how to explicitly compute the homology of modular curves using modular symbols.

We assume the reader is familiar with basic notions of algebraic topology, including homology groups of surfaces and triangulation. We also assume that the reader has read Chapter 5 about the fundamental domain for the action of PSL_{2}(Z) on the upper half plane and the analytic construction of modular curves.

Some standard references for modular symbols are [Man72] [Lan95, IV], [Cre97], and [Mer94]. Sections 7.1–7.2 below very closely follow Section 1 of Manin’s paper [Man72].

For the rest of this chapter, let Γ = PSL_{2}(Z) and let G be a subgroup of Γ of finite index. Note that we do not require G to be a congruence subgroup. The quotient X(G) = G\𝔥^{∗} of 𝔥^{∗} = 𝔥 ∪ P^{1}(Q) by G has an induced structure of a compact Riemann surface. Let π : 𝔥^{∗} → X(G) denote the natural projection. The matrices

s = (\begin{matrix}0 & − 1 \\ 1 & 0 \end{matrix}) and t = (\begin{matrix}1 & − 1 \\ 1 & 0 \end{matrix})

together generate Γ; they have orders 2 and 3, respectively.

7.1 Modular symbols

Let H^{0}(X(G), Ω^{1}) denote the complex vector space of holomorphic 1-forms on X(G). Integration of differentials along homology classes defines a perfect pairing

H_{1}(X(G), R) × H^{0}(X(G), Ω^{1}) → C,

hence an isomorphism

H_{1}(X(G), R) ≅ Hom_{C}(H^{0}(X(G), Ω^{1}), C).

For more details, see [Lan95, §IV.1].

Given two elements α, β ∈ 𝔥^{∗}, integration from α to β induces a well-defined element of Hom_{C}(H^{0}(X(G), Ω^{1}), C), hence an element

{α, β} ∈ H_{1}(X(G), R).

Definition 7.1.1 (Modular symbol). The homology class {α, β} ∈ H_{1}(X(G), R) associated to α, β ∈ 𝔥^{∗} is called the modular symbol attached to α and β.

Proposition 7.1.2. The symbols {α, β} have the following properties:

1. {α, α} = 0, {α, β} = −{β, α}, and {α, β} + {β, γ} + {γ, α} = 0.

2. {gα, gβ} = {α, β} for all g ∈ G

3. If X(G) has nonzero genus, then {α, β} ∈ H_{1}(X(G), Z) if and only if Gα = Gβ (i.e., we have π(α) = π(β)).

Remark 7.1.3. We only have {α, β} = {β, α} if {α, β} = 0, so the modular symbols notation, which suggests “unordered pairs,” is actively misleading.

Proposition 7.1.4. For any α ∈ 𝔥^{∗}, the map G → H_{1}(X(G), Z) that sends g to {α, gα} is a surjective group homomorphism that does not depend on the choice of α.

Proof. If g, h ∈ G and α ∈ 𝔥^{∗}, then

{α, gh(α)} = {α, gα} + {gα, ghα} = {α, gα} + {α, hα},

so the map is a group homomorphism. To see that the map does not depend on the choice of α, suppose β ∈ 𝔥^{∗}. By Proposition 7.1.2, we have {α, β} = {gα, gβ}. Thus

{α, gα} + {gα, β} = {gα, β} + {β, gβ},

so cancelling {gα, β} from both sides proves the claim.

The fact that the map is surjective follows from general facts from algebraic topology. Let 𝔥^{0} be the complement of Γi ∪ Γρ in 𝔥, fix α ∈ 𝔥^{0}, and let X(G)^{0} = π(𝔥^{0}). The map 𝔥^{0} → X(G)^{0} is an unramified covering of (noncompact) Riemann surfaces with automorphism group G. Thus α determines a group homomorphism π_{1}(X(G)^{0}, π(α)) → G. When composed with the morphism G → H_{1}(X(G), Z) above, the composition

π_{1}(X(G)^{0}, π(α)) → G → H_{1}(X(G), Z)

is the canonical map from the fundamental group of X(G)^{0} to the homology of the corresponding compact surface, which is surjective. This forces the map G → H_{1}(X(G), Z) to be surjective, which proves the claim.

7.2 Manin symbols

We continue to assume that G is a finite-index subgroup of Γ = PSL_{2}(Z), so the set G\Γ = {Gg_{1}, … Gg_{d}} of right cosets of G in Γ is finite. Manin symbols are a certain finite subset of modular symbols that are indexed by right cosets of G in Γ.

7.2.1 Using continued fractions to obtain surjectivity

Let R = G\Γ be the set of right cosets of G in Γ. Define

[ ] : R → H_{1}(X(G), R)

by [r] = {r0, r∞}, where r0 means the image of 0 under any element of the coset r (it doesn’t matter which). For g ∈ Γ, we also write [g] = [gG].

Proposition 7.2.1. Any element of H_{1}(X(G), Z) is a sum of elements of the form [r], and the representation ∑ n_{r}{α_{r}, β_{r}} of h ∈ H_{1}(X(G), Z) can be chosen so that ∑ n_{r}(π(β_{r}) − π(α_{r})) = 0 ∈ Div(X(G)).

Proof. By Proposition 7.1.4, every element h of H_{1}(X(G), Z) is of the form {0, g(0)} for some g ∈ G. If g(0) = ∞, then h = [G] and π(∞) = π(0), so we may assume g(0) = a/b ≠ ∞, with a/b in lowest terms and b > 0. Also assume a > 0, since the case a < 0 is treated in the same way. Let

0 = \frac{p_{−2}}{q_{−2}} = \frac{0}{1}, \frac{p_{−1}}{q_{−1}} = \frac{1}{0}, \frac{p_{0}}{1} = \frac{p_{0}}{q_{0}} , \frac{p_{1}}{q_{1}} , \frac{p_{2}}{q_{2}} , …, \frac{p_{n}}{q_{n}} = \frac{a}{b}

denote the continued fraction convergents of the rational number a/b. Then

p_{j}q_{j−1} − p_{j−1}q_{j} = (−1)^{j−1} for − 1 ≤ j ≤ n.

If we let g_{j} = (\begin{matrix}(−1)^{j−1}p_{j} & p_{j−1} \\ (−1)^{j−1}q_{j} & q_{j−1} \end{matrix}), then g_{j} ∈ SL_{2}(Z) and

\{0, \frac{a}{b}\} = ∑_{j=−1}^{n} \{\frac{p_{j−1}}{q_{j−1}} , \frac{p_{j}}{q_{j}} \}

= ∑_{j=−1}^{n}{g_{j}0, g_{j}∞})

= ∑_{j=−1}^{r}[g_{j}].

For the assertion about the divisor sum equaling zero, notice that the endpoints of the successive modular symbols cancel out, leaving the difference of 0 and g(0) in the divisor group, which is 0.

Lemma 7.2.2. If x = ∑_{j=1}^{t} n_{j}{α_{j}, β_{j}} is a Z-linear combination of modular symbols for G and ∑ n_{j}(π(β_{j}) − π(α_{j})) = 0 ∈ Div(X(G)), then x ∈ H_{1}(X(G), Z).

Proof. We may assume that each n_{j} is ±1 by allowing duplication. We may further assume that each n_{j} = 1 by using that {α, β} = −{β, α}. Next reorder the sum so π(β_{j}) = π(α_{j+1}) by using that the divisor is 0, so every β_{j} must be equivalent to some α_{j′}, etc. The lemma should now be clear.

56 7. Modular Symbols

FIGURE 7.2.1.

7.2.2 Triangulating X(G) to obtain injectivity

Let C be the abelian group generated by symbols (r) for r ∈ G\Γ, subject to the relations

(r) + (rs) = 0, and (r) = 0 if r = rs.

For (r) ∈ C, define the boundary of (r) to be the difference π(r∞) − π(r0) ∈ Div(X(G)). Since s swaps 0 and ∞, the boundary map is a well-defined map on C. Let Z be its kernel.

Let B be the subgroup of C generated by symbols (r), for all r ∈ G\Γ that satisfy r = rt, and by (r) + (rt) + (rt^{2}) for all other r. If r = rt, then rt(0) = r(0), so r(∞) = r(0), so (r) ∈ Z. Also, using (7.2.1) below, we see that for any r, the element (r) + (rt) + (rt^{2}) lies in Z.

The map G\Γ → H_{1}(X(G), R) that sends (r) to [r] induces a homomorphism C → H_{1}(X(G), R), so by Proposition 7.2.1 we obtain a surjective homomorphism

ψ : Z/B → H_{1}(X(G), Z).

Theorem 7.2.3 (Manin). The map ψ : Z/B → H_{1}(X(G), Z) is an isomorphism.

Proof. We only have to prove that ψ is injective. Our proof follows the proof of [Man72, Thm. 1.9] very closely. We compute the homology H_{1}(X(G), Z) by triangulating X(G) to obtain a simplicial complex L with homology Z_{1}/B_{1}, then embed Z/B in the homology Z_{1}/B_{1} of X(G). Most of our work is spent describing the triangulation L.

Let E denote the interior of the triangle with vertices 0, 1, and ∞, as illustrated in Figure 7.2.1. Let E′ denote the union of the interior of the region bounded by the path from i to ρ = e^{πi/3} to 1 + i to ∞ with the indicated path from i to ρ, not including the vertex i.

When reading the proof below, it will be helpful to look at the following table, which illustrates what s = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), t = ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ), and t^{2} do to the vertices in Figure 7.2.1:

\begin{matrix}1 & 0 & 1 & ∞ & i & 1 + i & (1 + i)/2 & ρ \\ s & ∞ & −1 & 0 & i & (−1 + i)/2 & −1 + i & −\overline{ρ} \\ t & ∞ & 0 & 1 & 1 + i & (1 + i)/2 & i & ρ \\ t^{2} & 1 & ∞ & 0 & (1 + i)/2 & i & 1 + i & ρ \end{matrix}  (7.2.1)

Note that each of E′, tE′, and t^{2}E′ is a fundamental domain for Γ, in the sense that every element of the upper half plane is conjugate to exactly one element in the closure of E′ (except for identifications along the boundaries). For example, E′ is obtained from the standard fundamental domain for Γ, which has vertices ρ^{2}, ρ, and ∞, by chopping it in half along the imaginary axis, and translating the piece on the left side horizontally by 1.

If (0, ∞) is the path from 0 to ∞, then t(0, ∞) = (∞, 1) and t^{2}(0, ∞) = (1, 0). Also, s(0, ∞) = (∞, 0). Thus each half side of E is Γ-conjugate to the side from i to ∞. Also, each 1-simplex in Figure 7.2.1, i.e., the sides that connected two adjacent labeled vertices such as i and ρ, maps homeomorphically into X(Γ). This is clear for the half sides, since they are conjugate to a path in the interior of the standard fundamental domain for Γ, and for the medians (lines from midpoints to ρ) since the path from i to ρ is on an edge of the standard fundamental domain with no self identifications.

We now describe our triangulation L of X(G):

0-cells The 0 cells are the cusps π(P^{1}(Q)) and i-elliptic points π(Γi). Note that these are the images under π of the vertices and midpoints of sides of the triangles gE, for all g ∈ Γ.

1-cells The 1 cells are the images of the half-sides of the triangles gE, for g ∈ Γ, oriented from the edge to the midpoint (i.e., from the cusp to the i-elliptic point). For example, if r = Gg is a right coset, then

e_{1}(r) = π(g(∞), g(i)) ∈ X(G)

is a 1 cell in L. Since, as we observed above, every half side is Γ-conjugate to e_{1}(G), it follows that every 1-cell is of the form e(r) for some right coset r ∈ G\Γ.

Next observe that if r ≠ r′ then

e_{1}(r) = e_{1}(r′)  implies  r′ = rs.  (7.2.2)

Indeed, if π(g(∞), g(i)) = π(g′(∞), g′(i)), then ri = r′i (note that the endpoints of a path are part of the definition of the path). Thus there exists h, h′ ∈ G such that hg(i) = h′g′(i). Since the only nontrivial element of Γ that stabilizes i is s, this implies that (hg)^{−1}h′g′ = s. Thus h′g′ = hgs, so Gg′ = Ggs, so r′ = rs.

2-cells There are two types of 2-cells, those with 2 sides and those with 3.

2-sided: The 2-sided 2-cells e_{2}(r) are indexed by the cosets r = Gg such that rt = r. Note that for such an r, we have π(rE′) = π(rtE′) = π(rt^{2}E′). The 2-cell e_{2}(r) is π(gE′). The image g(ρ, i) of the half median maps to a line from the center of e_{2}(r) to the edge π(g(i)) = π(g(1 + i)). Orient e_{2}(r) in a way compatible with the e_{1}. Since Ggt = Gg,

π(g(1 + i), g(∞)) = π(gt^{2}(1 + i), gt^{2}(∞)) = π(g(i), g(0)) = π(gs(i), gs(∞)),

so

e_{1}(r)−e_{1}(rs) = π(g(∞), g(i))+π(gs(i), gs(∞)) = π(g(∞), g(i))+π(g(1+i), g(∞)).

Thus

∂e_{2}(r) = e_{1}(r) − e_{1}(rs).

Finally, note that if r′ ≠ r also satisfies r′t = r′, then e_{2}(r) ≠ e_{2}(r′) (to see this use that E′ is a fundamental domain for Γ).

3-sided: The 3-sided 2-cells e_{2}(r) are indexed by the cosets r = Gg such that rt ≠ r. Note that for such an r, the three triangles rE′, rtE′, and rt^{2}E′ are distinct (since they are nontrivial translates of a fundamental domain). Orient e_{2}(r) in a way compatible with the e_{1} (so edges go from cusps to midpoints). Then

∂e_{2}(r) = ∑_{n=0}^{2} (e_{1}(rt^{n}) − e_{1}(rt^{n}s)) .

We have now defined a complex L that is a triangulation of X(G). Let C_{1}, Z_{1}, and B_{1} be the group of 1-chains, 1-cycles, and 1-boundaries of the complex L. Thus C_{1} is the abelian group generated by the paths e_{1}(r), the subgroup Z_{1} is the kernel of the map that sends e_{1}(r) = π(r(∞), r(0)) to π(r(0)) − π((∞)), and B_{1} is the subgroup of Z_{1} generated by boundaries of 2-cycles.

Let C, Z, B be as defined before the statement of the Theorem 7.2.3. We have H_{1}(X(G), Z) ≅ Z_{1}/B_{1}, and would like to prove that Z/B ≅ Z_{1}/B_{1}.

Define a map φ : C → C_{1} by (r) ↦ e_{1}(rs) − e_{1}(r). The map φ is well defined because if r = rs, then clearly (r) ↦ 0, and (r) + (rs) maps to e_{1}(rs) − e_{1}(r) + e_{1}(r) − e_{1}(rs) = 0. To see that f is injective, suppose ∑ n_{r}(r) ≠ 0. Since in C we have the relations (r) = −(rs) and (r) = 0 if rs = r, we may assume that n_{r}n_{rs} = 0 for all r. We have

φ (∑ n_{r}(r)) = ∑ n_{r}(e_{1}(rs) − e_{1}(r)).

If n_{r} ≠ 0 then r ≠ rs, so (7.2.2) implies that e_{1}(r) ≠ e_{1}(rs). If n_{r} ≠ 0 and n_{r′} ≠ 0 with r′ ≠ r, then r ≠ rs and r′ ≠ r′s, so e_{1}(r), e_{1}(rs), e_{1}(r′), e_{1}(r′s) are all distinct. We conclude that ∑ n_{r}(e_{1}(rs) − e_{1}(r)) ≠ 0, which proves that φ is injective.

Suppose (r) ∈ C. Then

φ(r) + B_{1} = ψ(r) = {r(0), r(∞)} ∈ H_{1}(X(G), Z) = C_{1}/B_{1},

since

φ(r) = e_{1}(rs)−e_{1}(r) = π(rs(∞), rs(i))−π(r(∞), r(i)) = π(r(0), r(i))−π(r(∞), r(i))

belongs to the homology class {r(0), r(∞)}. Extending linearly, we have, for any z ∈ C, that φ(z) + B_{1} = ψ(z).

The generators for B_{1} are the boundaries of 2-cells e_{2}(r). As we saw above, these have the form φ(r) for all r such that r = rt, and φ(r) + φ(rt) + φ(rt^{2}) for the r such that rt ≠ r. Thus B_{1} = φ(B) ⊂ φ(Z), so the map φ induces an injection Z/B ↪ Z_{1}/B_{1}. This completes the proof of the theorem.

7.3 Hecke operators

In this section we will only consider the modular curve X_{0}(N) associated to the subgroup Γ_{0}(N) of SL_{2}(Z) of matrices that are upper triangular modulo N. Much of what we say will also be true, possibly with slight modification, for X_{1}(N), but not for arbitrary finite-index subgroups.

There is a commutative ring

T = Z[T_{1}, T_{2}, T_{3}, …]

of Hecke operators that acts on H_{1}(X_{0}(N), R). We will frequently revisit this ring, which also acts on the Jacobian J_{0}(N) of X_{0}(N), and on modular forms. The ring T is generated by T_{p}, for p prime, and as a free Z-module T is isomorphic to Z^{g}, where g is the genus of X_{0}(N). We will not prove these facts here (see ^{1} ). 1 Suppose

1Add some references and pointers to other parts of this book.

{α, β} ∈ H_{1}(X_{0}(N), R),

is a modular symbol, with α, β ∈ P^{1}(Q). For g ∈ M_{2}(Z), write g({α, β}) = {g(α), g(β)}. This is not a well-defined action of M_{2}(Z) on H_{1}(X_{0}(N), R), since {α′, β′} = {α, β} ∈ H_{1}(X_{0}(N), R) does not imply that {g(α′), g(β′)} = {g(α), g(β)}. Example 7.3.1. Using Magma we see that the homology H_{1}(X_{0}(11), R) is generated by {−1/7, 0} and {−1/5, 0}.

> M := ModularSymbols(11);   // Homology relative to cusps,
                             // with Q coefficients.
> S := CuspidalSubspace(M);  // Homology, with Q coefficients.
> Basis(S);
[ {-1/7, 0}, {-1/5, 0} ]

Also, we have 5{0, ∞} = {−1/5, 0}.

> pi := ProjectionMap(S);    // The natural map M --> S.
> M.3;
{oo, 0}
> pi(M.3);
-1/5*{-1/5, 0}

Let g = ( \begin{matrix}2 & 0 \\ 0 & 1 \end{matrix} ). Then 5\{g(0), g(∞)\} is not equal to \{g(−1/5), g(0)\}, so g does not define a well-defined map on H_{1}(X_{0}(11), R).

> x := 5*pi(M!<1,[Cusps()|0,Infinity()]>);
> y := pi(M!<1,[-2/5,0]>);
> x;
{-1/5, 0}
> y;
-1*{-1/7, 0} + -1*{-1/5, 0}
> x eq y;
false

Definition 7.3.2 (Hecke operators). We define the Hecke operator T_{p} on H_{1}(X_{0}(N), R) as follows. When p is a prime with p ∤ N, we have

T_{p}(\{α, β\}) = (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) (\{α, β\}) + ∑_{r=0}^{p−1} (\begin{matrix}1 & r \\ 0 & p \end{matrix}) (\{α, β\}).

When p | N, the formula is the same, except that the first summand, which involves ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ), is omitted.

Example 7.3.3. We continue with Example 7.3.1. If we apply the Hecke operator T_{2} to both 5{0, ∞} and {−1/5, 0}, the “non-well-definedness” cancels out.

> x := 5*pi(M!<1,[Cusps()|0,Infinity()]> +
     M!<1,[Cusps()|0,Infinity()]> + M!<1,[Cusps()|1/2,Infinity()]>);
> x;
-2*{-1/5, 0}
> y := pi(M!<1,[-2/5,0]>+ M!<1,[-1/10,0]> + M!<1,[2/5,1/2]>);
> y;
-2*{-1/5, 0}

Examples 7.3.1 shows that it is not clear that the definition of T_{p} given above makes sense. For example, if {α, β} is replaced by an equivalent modular symbol {α′, β′}, why does the formula for T_{p} give the same answer? We will not address this question further here, but will revisit it later^{2} when we have a more natural 2 and intrinsic definition of Hecke operators. We only remark that T_{p} is induced by a “correspondence” from X_{0}(N) to X_{0}(N), so T_{p} preserve H_{1}(X_{0}(N), Z).

7.4 Modular symbols and rational homology

In this section we sketch a beautiful proof, due to Manin, of a result that is crucial to our understanding of rationality properties of special values of L-functions. For example, Mazur and Swinnerton-Dyer write in [MSD74, §6], “The modular symbol is essential for our theory of p-adic Mellin transforms,” right before discussing this rationality result. Also, as we will see in the next section, this result implies that if E is an elliptic curve over Q, then L(E, 1)/Ω_{E} ∈ Q, which confirms a consequence of the Birch and Swinnerton-Dyer conjecture.

Theorem 7.4.1 (Manin). For any α, β ∈ P^{1}(Q), we have

{α, β} ∈ H_{1}(X_{0}(N), Q).

Proof (sketch). Since {α, β} = {α, ∞} − {β, ∞}, it suffices to show that {α, ∞} ∈ H_{1}(X_{0}(N), Q) for all α ∈ Q. We content ourselves with proving that {0, ∞} ∈ H_{1}(X_{0}(N), Z), since the proof for general {0, α} is almost the same.

We will use that the eigenvalues of T_{p} on H_{1}(X_{0}(N), R) have absolute value bounded by 2\sqrt{p}, a fact that was proved by Weil (the Riemann hypothesis for curves over finite fields). Let p ∤ N be a prime. Then

T_{p}(\{0, ∞\}) = \{0, ∞\} + ∑_{r=0}^{p−1} \{\frac{r}{p}, ∞\} = (1 + p)\{0, ∞\} + ∑_{r=0}^{p−1} \{\frac{r}{p}, 0\} ,

so

(1 + p − T_{p})(\{0, ∞\}) = ∑_{r=0}^{p−1} \{0, \frac{r}{p}\} .

Since p ∤ N, the cusps 0 and r/p are equivalent (use the Euclidean algorithm to find a matrix in SL_{2}(Z) of the form ( \begin{matrix}r & ∗ \\ p & ∗ \end{matrix})), so the modular symbols \{0, r/p\}, for r = 0, 1, …, p − 1 all lie in H_{1}(X_{0}(N), Z). Since the eigenvalues of T_{p} have absolute value at most 2\sqrt{p}, the linear transformation 1 + p − T_{p} of H_{1}(X_{0}(N), Z) is invertible. It follows that some integer multiple of {0, ∞} lies in H_{1}(X_{0}(N), Z), as claimed.

There are general theorems about the denominator of {α, β} in some cases. Example 7.3.1 above demonstrated the following theorem in the case N = 11.

Theorem 7.4.2 (Ogg [Ogg71]). Let N be a prime. Then the image

[{0, ∞}] ∈ H_{1}(X_{0}(N), Q)/ H_{1}(X_{0}(N), Z)

has order equal to the numerator of (N − 1)/12.

2Say where, when I write this later.

7.5 Special values of L-functions 61

7.5 Special values of L-functions

This section is a preview of one of the central arithmetic results we will discuss in more generality later in this book.^{3} 3

The celebrated modularity theorem of Wiles et al. asserts that there is a correspondence between isogeny classes of elliptic curves E of conductor N and normalized new modular eigenforms f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) with a_{n} ∈ Z. This correspondence is characterized by the fact that for all primes p ∤ N, we have a_{p} = p + 1 − #E(F_{p}).

Recall that a modular form for Γ_{0}(N) of weight 2 is a holomorphic function f : 𝔥 → C that is “holomorphic at the cusps” and such that for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),

f (\frac{az + b}{cz + d}) = (cz + d)^{2}f(z).

Suppose E is an elliptic curve that corresponds to a modular form f. If L(E, s) is the L-function attached to E, then

L(E, s) = L(f, s) = ∑ \frac{a_{n}}{n^{s}} ,

so, by a theorem of Hecke which we will prove [later]^{4} , L(f, s) is holomorphic on 4 all C. Note that L(f, s) is the Mellin transform of the modular form f:

L(f, s) = (2π)^{s}Γ(s)^{−1} ∫_{0}^{i∞}(−iz)^{s}f(z)\frac{dz}{z} .  (7.5.1)

The Birch and Swinnerton-Dyer conjecture concerns the leading coefficient of the series expansion of L(E, s) about s = 1. A special case is that if L(E, 1) ≠ 0, then

\frac{L(E, 1)}{Ω_{E}} = \frac{∏ c_{p} · #� (E)}{#E(Q)_{tor}^{2}}.

Here Ω_{E} = | ∫_{E(R)} ω|, where ω is a “Néron” differential 1-form on E, i.e., a generator for H^{0}(E, Ω_{E/Z}^{1}), where E is the Néron model of E. (The Néron model of E is the unique, up to unique isomorphism, smooth group scheme E over Z, with generic fiber E, such that for all smooth schemes S over Z, the natural map Hom_{Z}(S, E) → Hom_{Q}(S × Spec(Q), E) is an isomorphism.) In particular, the conjecture asserts that for any elliptic curve E we have L(E, 1)/Ω_{E} ∈ Q.

Theorem 7.5.1. Let E be an elliptic curve over Q. Then L(E, 1)/Ω_{E} ∈ Q.

Proof (sketch). By the modularity theorem of Wiles et al., E is modular, so there is a surjective morphism π_{E} : X_{0}(N) → E, where N is the conductor of E. This implies that there is a newform f that corresponds to (the isogeny class of) E, with L(f, s) = L(E, s). Also assume, without loss of generality, that E is “optimal” in its isogeny class, which means that if X_{0}(N) → E′ → E is a sequence of morphism whose composition is π_{E} and E′ is an elliptic curve, then E′ = E.

3Where?

4where?

By Equation 7.5.1, we have

L(E, 1) = 2π ∫_{0}^{i∞} −izf(z)dz/z.  (7.5.2)

If q = e^{2πiz}, then dq = 2πiqdz, so 2πif(z)dz = dq/q, and (7.5.2) becomes

L(E, 1) = − ∫_{0}^{i∞} f(q)dq.

Recall that Ω_{E} = | ∫_{E(R)} ω|, where ω is a Néron differential on E. The expression f(q)dq defines a differential on the modular curve X_{0}(N), and there is a rational number c, the Manin constant, such that π_{E}^{∗}ω = cf(q)dq. More is true: Edixhoven proved (as did Ofer Gabber) that c ∈ Z; also Manin conjectured that c = 1 and Edixhoven proved (unpublished) that if p | c, then p = 2, 3, 5, 7.

A standard fact is that if

L = {∫_{γ} ω : γ ∈ H_{1}(E, Z)}

is the period lattice of E associated to ω, then E(C) ≅ C/L. Note that Ω_{E} is either the least positive real element of L or twice this least positive element (if E(R) has two real components).

The next crucial observation is that by Theorem 7.4.1, there is an integer n such that n{0, ∞} ∈ H_{1}(X_{0}(N), Z). This is relevant because if

L′ = {∫_{γ} f(q)dq : γ ∈ H_{1}(X_{0}(N), Z)} ⊂ C.

then L = \frac{1}{c}L′ ⊂ L′. This assertion follows from our hypothesis that E is optimal and standard facts about complex tori and Jacobians, which we will prove later [in this course/book].

One can show that L(E, 1) ∈ R, for example, by writing down an explicit real convergent series that converges to L(E, 1). This series is used in algorithms to compute L(E, 1), and the derivation of the series uses properties of modular forms that we have not yet developed. Another approach is to use complex conjugation to define an involution ∗ on H_{1}(X_{0}(N), R), then observe that {0, ∞} is fixed by ∗. (The involution ∗ is given on modular symbols by ∗{α, β} = {−α, −β}.)

Since L(E, 1) ∈ R, the integral

∫_{n{0,∞}} f(q)dq = n ∫_{0}^{i∞} f(q)dq = −nL(E, 1) ∈ L′

lies in the subgroup (L′)^{+} of elements fixed by complex conjugation. If c is the Manin constant, we have cnL(E, 1) ∈ L^{+}. Since Ω_{E} is the least nonzero element of L^{+} (or twice it), it follows that 2cnL(E, 1)/Ω_{E} ∈ Z, which proves the proposition.

Modular Forms of Higher Level

8.1 Modular Forms on Γ_{1}(N)

Fix integers k ≥ 0 and N ≥ 1. Recall that Γ_{1}(N) is the subgroup of elements of SL_{2}(Z) that are of the form ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) when reduced modulo N.

Definition 8.1.1 (Modular Forms). The space of modular forms of level N and weight k is

M_{k}(Γ_{1}(N)) = {f : f(γτ) = (cτ + d)^{k}f(τ) all γ ∈ Γ_{1}(N)} ,

where the f are assumed holomorphic on 𝔥 ∪ {cusps} (see below for the precise meaning of this). The space of cusp forms of level N and weight k is the subspace S_{k}(Γ_{1}(N)) of M_{k}(Γ_{1}(N)) of modular forms that vanish at all cusps.

Remark 8.1.2. In the beginning of this book (e.g., Section 1.3) we often wrote S_{k}(N) for S_{k}(Γ_{1}(N)). Since there are many congruence subgroups, to avoid confusion we will write out S_{k}(Γ_{1}(N)) in the rest of this book.

Suppose f ∈ M_{k}(Γ_{1}(N)). The group Γ_{1}(N) contains the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), so

f(z + 1) = f(z),

and for f to be holomorphic at infinity means that f has a Fourier expansion

f = ∑_{n=0}^{∞} a_{n}q^{n}.

To explain what it means for f to be holomorphic at all cusps, we introduce some additional notation. For α ∈ GL_{2}^{+}(R) and f : 𝔥 → C define another function f_{|[α]_{k}} as follows:

f_{|[α]_{k}}(z) = det(α)^{k−1}(cz + d)^{−k}f(αz).

It is straightforward to check that f_{|[αα′]_{k}} = (f_{|[α]_{k}})_{|[α′]_{k}}. Note that we do not have to make sense of f_{|[α]_{k}}(∞), since we only assume that f is a function on 𝔥 and not 𝔥^{∗}.

Using our new notation, the transformation condition required for f : 𝔥 → C to be a modular form for Γ_{1}(N) of weight k is simply that f be fixed by the [ ]_{k}- action of Γ_{1}(N). Suppose x ∈ P^{1}(Q) is a cusp, and choose α ∈ SL_{2}(Z) such that α(∞) = x. Then g = f_{|[α]_{k}} is fixed by the [ ]_{k} action of α^{−1}Γ_{1}(N)α.

Lemma 8.1.3. Let α ∈ SL_{2}(Z). Then there exists a positive integer h such that

( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ) ∈ α^{−1}Γ_{1}(N)α.

Proof. This follows from the general fact that the set of congruence subgroups of SL_{2}(Z) is closed under conjugation by elements α ∈ SL_{2}(Z), and every congruence subgroup contains an element of the form ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ). If G is a congruence subgroup, then ( \begin{matrix}1 & N \\ 0 & 1 \end{matrix} ) ∈ Γ(N) ⊂ G for some N, and α^{−1}Γ(N)α = Γ(N), since Γ(N) is normal, so Γ(N) ⊂ α^{−1}Gα.

Letting h be as in the lemma, we have g(z + h) = g(z). Then the condition that f be holomorphic at the cusp x is that

g(z) = ∑_{n≥0} b_{n/h}q^{1/h}

on the upper half plane. We say that f vanishes at x if b_{0} = 0, and a cusp form is a form that vanishes at every cusp.

8.2 Diamond bracket and Hecke operators

In this section we consider the spaces of modular forms S_{k}(Γ_{1}(N), ε), for Dirichlet characters ε mod N, and explicitly describe the action of the Hecke operators on these spaces.

8.2.1 Diamond bracket operators

The group Γ_{1}(N) is a normal subgroup of Γ_{0}(N), and the quotient Γ_{0}(N)/Γ_{1}(N) is isomorphic to (Z/NZ)^{∗}. From this structure we obtain an action of (Z/NZ)^{∗} on S_{k}(Γ_{1}(N)), and use it to decompose S_{k}(Γ_{1}(N)) as a direct sum of more manageable chunks S_{k}(Γ_{1}(N), ε).

Definition 8.2.1 (Dirichlet character). A Dirichlet character ε modulo N is a homomorphism

ε : (Z/NZ)^{∗} → C^{∗}.

We extend ε to a map ε : Z → C by setting ε(m) = 0 if (m, N) ≠ 1 and ε(m) = ε(m mod N) otherwise. If ε : Z → C is a Dirichlet character, the conductor of ε is the smallest positive integer c such that ε arises from a homomorphism (Z/cZ)^{∗} → C^{∗}.

Remarks 8.2.2.

1. If ε is a Dirichlet character modulo N and M is a multiple of N then ε induces a Dirichlet character mod M. If M is a divisor of N then ε is induced by a Dirichlet character modulo M if and only if M divides the conductor of ε.

2. The set of Dirichlet characters forms a group, which is non-canonically isomorphic to (Z/NZ)^{∗} (it is the dual of this group).

3. The mod N Dirichlet characters all take values in Q(e^{2πi/e}) where e is the exponent of (Z/NZ)^{∗}. When N is an odd prime power, the group (Z/NZ)^{∗} is cyclic, so e = φ(φ(N)). This double-φ can sometimes cause confusion.

4. There are many ways to represent Dirichlet characters with a computer (see, e.g., [Ste07a, Ch. 4]). One way is to fix generators for (Z/NZ)^{∗} in any way you like and represent ε by the images of each of these generators. Assume for the moment that N is odd. To make the representation more “canonical”, reduce to the prime power case by writing (Z/NZ)^{∗} as a product of cyclic groups corresponding to prime divisors of N. A “canonical” generator for (Z/p^{r}Z)^{∗} is then the smallest positive integer s such that s mod p^{r} generates (Z/p^{r}Z)^{∗}. Store the character that sends s to e^{2πin/φ(φ(p^{r}))} by storing the integer n. For general N, store the list of integers n_{p}, one p for each prime divisor of N (unless p = 2, in which case you store two integers n_{2} and n′_{2}, where n_{2} ∈ {0, 1}).

Definition 8.2.3. Let \overline{d} ∈ (Z/NZ)^{∗} and f ∈ S_{k}(Γ_{1}(N)). By Lemma 5.2.6, the map SL_{2}(Z) → SL_{2}(Z/NZ) is surjective, so there exists a matrix γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) such that d ≡ \overline{d} (mod N). The diamond bracket d operator is then

f(τ)|⟨d⟩ = f_{|[γ]_{k}} = f(γτ)(cτ + d)^{−k}.

The definition of ⟨d⟩ does not depend on the choice of lift matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ), since any two lifts differ by an element of Γ(N) and f is fixed by Γ(N) since it is fixed by Γ_{1}(N).

For each Dirichlet character ε mod N let

S_{k}(Γ_{1}(N), ε) = {f : f|⟨d⟩ = ε(d)f all d ∈ (Z/NZ)^{∗}}
= {f : f_{|[γ]_{k}} = ε(d_{γ})f all γ ∈ Γ_{0}(N)},

where d_{γ} is the lower-right entry of γ.

When f ∈ S_{k}(Γ_{1}(N), ε), we say that f has Dirichlet character ε. In the literature, sometimes f is said to be of “nebentypus” ε.

Lemma 8.2.4. The operator ⟨d⟩ on the finite-dimensional vector space S_{k}(Γ_{1}(N)) is diagonalizable.

Proof. There exists n ≥ 1 such that I = ⟨1⟩ = ⟨d^{n}⟩ = ⟨d⟩^{n}, so the characteristic polynomial of ⟨d⟩ divides the square-free polynomial X^{n} − 1.

Note that S_{k}(Γ_{1}(N), ε) is the ε(d) eigenspace of ⟨d⟩. Thus we have a direct sum decomposition

S_{k}(Γ_{1}(N)) = ⨁_{ε:(Z/NZ)^{∗}→C^{∗}} S_{k}(Γ_{1}(N), ε).

We have ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∈ Γ_{0}(N), so if f ∈ S_{k}(Γ_{1}(N), ε), then

f(τ)(−1)^{−k} = ε(−1)f(τ).

Thus S_{k}(Γ_{1}(N), ε) = 0, unless ε(−1) = (−1)^{k}, so about half of the direct summands S_{k}(Γ_{1}(N), ε) vanish.

8.2.2 Hecke operators on q-expansions

Suppose

f = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{k}(Γ_{1}(N), ε),

and let p be a prime. Then

f|T_{p} = \begin{cases}∑_{n=1}^{∞} a_{np}q^{n} + p^{k−1}ε(p) ∑_{n=1}^{∞} a_{n}q^{pn}, & p ∤ N \\ ∑_{n=1}^{∞} a_{np}q^{n} + 0. & p | N. \end{cases}

Note that ε(p) = 0 when p | N, so the second part of the formula is redundant.

When p | N, T_{p} is often denoted U_{p} in the literature, but we will not do so here. Also, the ring T generated by the Hecke operators is commutative, so it is harmless, though potentially confusing, to write T_{p}(f) instead of f|T_{p}.

We record the relations

T_{m}T_{n} = T_{mn}, (m, n) = 1,
T_{p^{k}} = \begin{cases}(T_{p})^{k}, & p | N \\ T_{p^{k−1}}T_{p} − ε(p)p^{k−1}T_{p^{k−2}}, & p ∤ N. \end{cases}

WARNING: When p | N, the operator T_{p} on S_{k}(Γ_{1}(N), ε) need not be diagonalizable.

8.3 Old and new subspaces

Let M and N be positive integers such that M | N and let t | \frac{N}{M}. If f(τ) ∈ S_{k}(Γ_{1}(M)) then f(tτ) ∈ S_{k}(Γ_{1}(N)). We thus have maps

S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N))

for each divisor t | \frac{N}{M}. Combining these gives a map

φ_{M} : ⨁_{t|(N/M)} S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N)).

Definition 8.3.1 (Old Subspace). The old subspace of S_{k}(Γ_{1}(N)) is the subspace generated by the images of the φ_{M} for all M | N with M ≠ N.

Definition 8.3.2 (New Subspace). The new subspace of S_{k}(Γ_{1}(N)) is the complement of the old subspace with respect to the Petersson inner product (see Section 4.6).

[[TODO: We only defined the Petersson inner product in Section 4.6 in the case when N = 1.]]

Definition 8.3.3 (Newform). A newform is an element f of the new subspace of S_{k}(Γ_{1}(N)) that is an eigenvector for every Hecke operator, which is normalized so that the coefficient of q in f is 1.

If f = ∑ a_{n}q^{n} is a newform then the coefficients a_{n} are algebraic integers, which have deep arithmetic significance. For example, when f has weight 2, there is an associated abelian variety A_{f} over Q of dimension [Q(a_{1}, a_{2}, …) : Q] such that ∏ L(f^{σ}, s) = L(A_{f}, s), where the product is over the Gal(\overline{Q}/Q)-conjugates of f. The abelian variety A_{f} was constructed by Shimura as follows. Let J_{1}(N) be the Jacobian of the modular curve X_{1}(N). As we will see, the ring T of Hecke operators acts naturally on J_{1}(N). Let I_{f} be the kernel of the homomorphism T → Z[a_{1}, a_{2}, …] that sends T_{n} to a_{n}. Then

A_{f} = J_{1}(N)/I_{f}J_{1}(N).

In the converse direction, it is a deep theorem of Breuil, Conrad, Diamond, Taylor, and Wiles that if E is any elliptic curve over Q, then E is isogenous to A_{f} for some f of level equal to the conductor N of E. More generally, building on this work, it is now known that if A is a simple abelian variety over Q with End(A) ⊗ Q a number field of degree dim(A), then A is isogenous to A_{f} for some newform f (see [Rib92, KW08]).

When f has weight greater than 2, in [Sch90] Scholl constructs, in an analogous way, a Grothendieck motive M_{f} attached to f.

68 8. Modular Forms of Higher Level

Newforms and Euler Products

In this chapter we discuss the work of Atkin, Lehner, and W. Li on newforms and their associated L-series and Euler products. Then we discuss explicitly how U_{p}, for p | N, acts on old forms, and how U_{p} can fail to be diagonalizable. Then we describe a canonical generator for S_{k}(Γ_{1}(N)) as a free module over T_{C}. Finally, we observe that the subalgebra of T_{Q} generated by Hecke operators T_{n} with (n, N) = 1 is isomorphic to a product of number fields.

9.1 Atkin-Lehner-Li theory

The results of [Li75] about newforms are proved using many linear transformations that do not necessarily preserve S_{k}(Γ_{1}(N), ε). Thus we introduce more general spaces of cusp forms, which these transformations preserve. These spaces are also useful because they make precise how the space of cusp forms for the principal congruence subgroup

Γ(N) = ker(SL_{2}(Z) → SL_{2}(Z/NZ))

can be understood in terms of spaces S_{k}(Γ_{1}(M), ε) for various M and ε, which justifies our usual focus on these latter spaces. This section follows [Li75] closely.

Let M and N be positive integers and define

Γ_{0}(M, N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z) : M | c, N | b\} ,

and

Γ(M, N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(M, N) : a ≡ d ≡ 1 (mod MN)\} .

Note that Γ_{0}(M, 1) = Γ_{0}(M) and Γ(M, 1) = Γ_{1}(M). Let S_{k}(M, N) denote the space of cusp forms for Γ(M, N).

If ε is a Dirichlet character modulo MN such that ε(−1) = (−1)^{k}, let S_{k}(M, N, ε) denote the space of all cups forms for Γ(M, N) of weight k and character ε. This is the space of holomorphic functions f : 𝔥 → C that satisfy the usual vanishing conditions at the cusps and such that for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(M, N),

f| (\begin{matrix}a & b \\ c & d \end{matrix}) = ε(d)f.

We have

S_{k}(M, N) = ⊕_{ε}S_{k}(M, N, ε).

We now introduce operators between various S_{k}(M, N). Note that, except when otherwise noted, the notation we use for these operators below is as in [Li75], which conflicts with notation in various other books. When in doubt, check the definitions.

Let

f| (\begin{matrix}a & b \\ c & d \end{matrix}) (τ) = (ad − bc)^{k/2}(cτ + d)^{−k}f (\frac{aτ + b}{cτ + d}) .

This is like before, but we omit the weight k from the bar notation, since k will be fixed for the whole discussion.

For any d and f ∈ S_{k}(M, N, ε), define

f|U_{d}^{N} = d^{k/2−1}f| ( ∑_{u mod d} (\begin{matrix}1 & uN \\ 0 & d \end{matrix})) ,

where the sum is over any set u of representatives for the integers modulo d. Note that the N in the notation U_{d}^{N} is a superscript, not a power of N. Also, let

f|B_{d} = d^{−k/2}f| (\begin{matrix}d & 0 \\ 0 & 1 \end{matrix}) ,

and

f|C_{d} = d^{k/2}f| (\begin{matrix}1 & 0 \\ 0 & d \end{matrix}) .

In [Li75], C_{d} is denoted W_{d}, which would be confusing, since in the literature W_{d} is usually used to denote a completely different operator (the Atkin-Lehner operator, which is denoted V_{d}^{M} in [Li75]).

Since ( \begin{matrix}1 & N \\ 0 & 1 \end{matrix} ) ∈ Γ(M, N), any f ∈ S_{k}(M, N, ε) has a Fourier expansion in terms of powers of q_{N} = q^{1/N}. We have

(∑_{n≥1} a_{n}q_{N}^{n} ) |U_{d}^{N} = ∑_{n≥1} a_{nd}q_{N}^{n},

(∑_{n≥1} a_{n}q_{N}^{n} ) |B_{d} = ∑_{n≥1} a_{n}q_{N}^{nd},

and

(∑_{n≥1} a_{n}q_{N}^{n} ) |C_{d} = ∑_{n≥1} a_{n}q_{Nd}^{n}.

The second two equalities are easy to see; for the first, write everything out and use that for n ≥ 1, the sum ∑_{u} e^{2πiun/d} is 0 or d if d ∤ n, d | n, respectively, much as in Section 3.5.2.

The maps B_{d} and C_{d} define injective maps between various spaces S_{k}(M, N, ε). To understand B_{d}, use the matrix relation

(\begin{matrix}d & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}x & y \\ z & w \end{matrix}) = (\begin{matrix}x & dy \\ z/d & w \end{matrix}) (\begin{matrix}d & 0 \\ 0 & 1 \end{matrix}) ,

and for C_{d} use

(\begin{matrix}1 & 0 \\ 0 & d \end{matrix}) (\begin{matrix}x & y \\ z & w \end{matrix}) = (\begin{matrix}x & y/d \\ zd & w \end{matrix}) (\begin{matrix}1 & 0 \\ 0 & d \end{matrix}) .

If d | N then B_{d} : S_{k}(M, N, ε) → S_{k}(dM, N/d, ε) is an isomorphism, and if d | M, then C_{d} : S_{k}(M, N) → S_{k}(M/d, Nd, ε) is also an isomorphism. In particular, taking d = N, we obtain an isomorphism

B_{N} : S_{k}(M, N, ε) → S_{k}(MN, 1, ε) = S_{k}(Γ_{1}(MN), ε).  (9.1.1)

Putting these maps together allows us to completely understand the cusp forms S_{k}(Γ(N)) in terms of spaces S_{k}(Γ_{1}(N^{2}), ε), for all Dirichlet characters ε that arise from characters modulo N. This is because S_{k}(Γ(N)) is isomorphic to the direct sum of S_{k}(N, N, ε), as ε various over all Dirichlet characters modulo N.

For any prime p, we define the pth Hecke operator on S_{k}(M, N, ε) by

T_{p} = U_{p}^{N} + ε(p)p^{k−1}B_{p}.

Note that T_{p} = U_{p}^{N} when p | N, since then ε(p) = 0. In terms of Fourier expansions, we have

(∑_{n≥1} a_{n}q_{N}^{n} ) |T_{p} = ∑_{n≥1} (a_{np} + ε(p)p^{k−1}a_{n/p}) q_{N}^{n},

where a_{n/p} = 0 if p ∤ n.

The operators we have just defined satisfy several commutativity relations. Suppose p and q are prime. Then T_{p}B_{q} = B_{q}T_{p}, T_{p}C_{q} = C_{q}T_{p}, and T_{p}U_{q}^{N} = U_{q}^{N}T_{p} if (p, qMN) = 1. Moreover U_{d}^{N}B_{d′} = B_{d′}U_{d}^{N} if (d, d′) = 1. Remark 9.1.1. Because of these relations, (9.1.1) describes S_{k}(Γ(N)) as a module over the ring generated by T_{p} for p ∤ N.

Definition 9.1.2 (Old Subspace). The old subspace S_{k}(M, N, ε)_{old} is the subspace of S_{k}(M, N, ε) generated by all f|B_{d} and g|C_{e} where f ∈ S_{k}(M′, N), g ∈ S_{k}(M, N′), and M′, N′ are proper factors of M, N, respectively, and d | M/M′, e | N/N′.

Since T_{p} commutes with B_{d} and C_{e}, the Hecke operators T_{p} preserve S_{k}(M, N, ε)_{old}, for p ∤ MN. Also, B_{N} defines an isomorphism

S_{k}(M, N, ε)_{old} ≅ S_{k}(MN, 1, ε)_{old}.

Definition 9.1.3 (Petersson Inner Product). If f, g ∈ S_{k}(Γ(N)), the Petersson inner product of f and g is

⟨f, g⟩ = \frac{1}{[SL_{2}(Z) : Γ(N)]} ∫_{D} f(z)\overline{g(z)}y^{k−2} dx dy,

where D is a fundamental domain for Γ(N) and z = x + iy.

This Petersson pairing is normalized so that if we consider f and g as elements of Γ(N′) for some multiple N′ of N, then the resulting pairing is the same (since the volume of the fundamental domain scales by the index).

Theorem 9.1.4 (Petersson). If p ∤ N and f ∈ S_{k}(Γ_{1}(N), ε), then ⟨f|T_{p}, g⟩ = ε(p)⟨f, g|T_{p}⟩.

See [Lan95, §VII.5, Thm. 5.1] for a proof of Theorem 9.1.4.

Remark 9.1.5. Theorem 9.1.4 implies that when p ∤ N the adjoint of T_{p} is ε(p)T_{p}, so T_{p} commutes with its adjoint, hence T_{p} is normal, which implies that T_{p} is diagonalizable. Be careful, because the T_{p}, with p | N, need not be diagonalizable (see Section 9.2.3).

Definition 9.1.6 (New Subspace). The new subspace S_{k}(M, N, ε)_{new} is the orthogonal complement of S_{k}(M, N, ε)_{old} in S_{k}(M, N, ε) with respect to the Petersson inner product.

Both the old and new subspaces of S_{k}(M, N, ε) are preserved by the Hecke operators T_{p} with p ∤ NM.

Remark 9.1.7. Li [Li75] also gives a purely algebraic definition of the new subspace as the intersection of the kernels of various trace maps from S_{k}(M, N, ε) that are obtained by averaging over coset representatives.

Definition 9.1.8 (Newform). A newform f = ∑ a_{n}q_{N}^{n} ∈ S_{k}(M, N, ε) is an element of S_{k}(M, N, ε)_{new} that is an eigenform for all T_{p}, for p ∤ NM, and is normalized so that a_{1} = 1.

Li introduces the crucial “Atkin-Lehner operator” W_{q}^{M} (denoted V_{q}^{M} in [Li75]), which plays a key roll in all the proofs, and is defined as follows. For a positive integer M and prime q, let α = ord_{q}(M) and use the extended Euclidean algorithm to find a choice of integers x, y, z such that q^{2α}x − yMz = q^{α}. Then W_{q}^{M} is

the operator defined by slashing with the matrix (\begin{matrix}q^{α}x & y \\ Mz & q^{α} \end{matrix}). Li shows that if

f ∈ S_{k}(M, 1, ε), then f|W_{q}^{M}|W_{q}^{M} = ε(q^{α})f, so W_{q}^{M} is invertible. Care must be taken, because the operator W_{q}^{M} need not commute with T_{p} = U_{p}^{N}, when p | M.

After proving many technical but elementary lemmas about the operators B_{d}, C_{d}, U_{p}^{N}, T_{p}, and W_{q}^{M}, Li uses the lemmas to deduce the following theorems, whose proofs are relatively elementary.

Theorem 9.1.9. Suppose f = ∑ a_{n}q_{N}^{n} ∈ S_{k}(M, N, ε) and a_{n} = 0 for all n with gcd(n, r) = 1, where r is a fixed positive integer. Then f ∈ S_{k}(M, N, ε)_{old}.

From the theorem we see that if f and g are newforms in S_{k}(M, N, ε), and if for all but finitely many primes p, the T_{p} eigenvalues of f and g are the same, then f − g is an old form, so f − g = 0, hence f = g. Thus the eigenspaces in the new subspace corresponding to the systems of Hecke eigenvalues associated to the T_{p}, with p ∤ MN, each have dimension 1. This is known as multiplicity one.

Theorem 9.1.10. Let f = ∑ a_{n}q_{N}^{n} be a newform in S_{k}(M, N, ε), p a prime with p ∤ MN, and q | MN a prime. Then

9.2 The U_{p} operator 73

1. f|T_{p} = a_{p}f, f|U_{q}^{N} = a_{q}f, and for all n ≥ 1,

a_{p}a_{n} = a_{np} + ε(p)p^{k−1}a_{n/p},
a_{q}a_{n} = a_{nq}.

If L(f, s) = ∑_{n≥1} a_{n}n^{−s} is the Dirichlet series associated to f, then L(f, s) has an Euler product

L(f, s) = ∏_{primes p}(1 − a_{p}p^{−s} + ε(p)p^{k−1}p^{−2s})^{−1}.

Note that when p | NM, we have 1 − a_{p}p^{−s} + ε(p)p^{k−1}p^{−2s} = 1 − a_{p}p^{−s}.

2. (a) If ε is not induced by a character mod MN/q, then |a_{q}| = q^{(k−1)/2}.

(b) If ε is induced by a character mod MN/q, then a_{q} = 0 if q^{2} | MN, and a_{q}^{2} = ε(q)q^{k−2} if q^{2} ∤ MN.

9.2 The U_{p} operator

Let N be a positive integer and M a divisor of N. For each divisor d of N/M we define a map

α_{d} : S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N)) : f(τ) ↦ f(dτ).

Thus α_{d} is just the map B_{d} from Section 26.4.2. We verify that f(dτ) ∈ S_{k}(Γ_{1}(N)) as follows. Recall that for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ), we write

(f|[γ]_{k})(τ) = det(γ)^{k−1}(cz + d)^{−k}f(γ(τ)).

The transformation condition for f to be in S_{k}(Γ_{1}(N)) is that f|[γ]_{k}(τ) = f(τ). Let f(τ) ∈ S_{k}(Γ_{1}(M)) and let ι_{d} = ( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ). Then f|[ι_{d}]_{k}(τ) = d^{k−1}f(dτ) is a modular form on Γ_{1}(N) since ι_{d}^{−1}Γ_{1}(M)ι_{d} contains Γ_{1}(N). Moreover, if f is a cusp form then so is f|[ι_{d}]_{k}.

Proposition 9.2.1. If f ∈ S_{k}(Γ_{1}(M)) is nonzero, then the images

\{α_{d}(f) : d | \frac{N}{M} \}

are linearly independent.

Proof. If the q-expansion of f is ∑ a_{n}q^{n}, then the q-expansion of α_{d}(f) is ∑ a_{n}q^{dn}. The matrix of coefficients of the q-expansions of α_{d}(f), for d | (N/M), is upper triangular. Thus the q-expansions of the α_{d}(f) are linearly independent, hence the α_{d}(f) are linearly independent, since the map that sends a cusp form to its q-expansion is linear.

When p | N, we denote by U_{p} the Hecke operator T_{p} acting on the space S_{k}(Γ_{1}(N)). For clarity, in this section only we will denote by T_{p}^{M}, the Hecke operator T_{p} ∈ End(S_{k}(Γ_{1}(M))). For f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{1}(N)), we have

f|U_{p} = ∑ a_{np}q^{n}.

Suppose f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{1}(M)) is a normalized eigenform for all of the Hecke operators T_{n} and ⟨n⟩, and p is a prime that does not divide M. Then

f|T_{p}^{M} = a_{p}f and f|⟨p⟩ = ε(p)f.

Assume N = p^{r}M, where r ≥ 1 is an integer. Let

f_{i}(τ) = f(p^{i}τ) = α_{p^{i}}(f),

so f_{0}, …, f_{r} are the images of f under the maps α_{p^{0}}, …, α_{p^{r}}, respectively, and f = f_{0}. We have

f|T_{p}^{M} = ∑_{n≥1} a_{np}q^{n} + ε(p)p^{k−1} ∑ a_{n}q^{pn}
= f_{0}|U_{p} + ε(p)p^{k−1}f_{1},

so

f_{0}|U_{p} = f|T_{p}^{M} − ε(p)p^{k−1}f_{1} = a_{p}f_{0} − ε(p)p^{k−1}f_{1}.  (9.2.1)

Also

f_{1}|U_{p} = (∑ a_{n}q^{pn}) |U_{p} = ∑ a_{n}q^{n} = f_{0}.

More generally, for any i ≥ 1, we have f_{i}|U_{p} = f_{i−1}.

The operator U_{p} preserves the two dimensional vector space spanned by f_{0} and f_{1}, and the matrix of U_{p} with respect to the basis f_{0}, f_{1} is

A = ( \begin{matrix}a_{p} & 1 \\ − ε(p)p^{k−1} & 0 \end{matrix}) ,

which has characteristic polynomial

X^{2} − a_{p}X + p^{k−1}ε(p).  (9.2.2)

When r ≥ 3, the operator U_{p} does not act diagonalizably on the space spanned by f_{0}, f_{1}, …, f_{r}. See Section 9.2.3 below.

9.2.1 A Connection with Galois representations

Equation (9.2.2) leads to a striking connection with Galois representations. Let f be a newform and let K = K_{f} be the field generated over Q by the Fourier coefficients of f. Let ℓ be a prime and λ a prime of the ring of integers of K lying over ℓ. Then, as we mentioned in Section 2.1, Deligne (and Serre, when k = 1, and Shimura when k = 2) constructed a representation

ρ_{f,λ} = ρ_{λ} : Gal(\overline{Q}/Q) → GL(2, K_{λ}).

If p ∤ Nℓ, then ρ_{λ} is unramified at p, so if Frob_{p} ∈ Gal(\overline{Q}/Q) if a Frobenius element, then ρ_{λ}(Frob_{p}) is well defined, up to conjugation. Moreover,

det(ρ_{λ}(Frob_{p})) = p^{k−1}ε(p), and
tr(ρ_{λ}(Frob_{p})) = a_{p}.

Thus the characteristic polynomial of ρ_{λ}(Frob_{p}) ∈ GL_{2}(K_{λ}) is

X^{2} − a_{p}X + p^{k−1}ε(p),

which is the same as (9.2.2).

9.2.2 When is U_{p} semisimple?

Question 9.2.2. Is U_{p} semisimple on the span of f_{0} and f_{1}?

If the eigenvalues of U_{p} acting on the span of f_{0} and f_{1} are distinct, then the answer is yes. If the eigenvalues are the same, then X^{2} − a_{p}X + p^{k−1}ε(p) has discriminant 0, so a_{p}^{2} = 4p^{k−1}ε(p), hence

a_{p} = 2p^{\frac{k−1}{2}} \sqrt{ε(p)}.

Open Problem 9.2.3. Does there exist an eigenform f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{1}(N)) such that a_{p} = 2p^{\frac{k−1}{2}} \sqrt{ε(p)}?

It is a curious fact that the Ramanujan conjectures, which were proved by Deligne in 1973, imply that |a_{p}| ≤ 2p^{(k−1)/2}, so the above equality remains taunting. When k = 2, Coleman and Edixhoven proved in [CE98] that |a_{p}| < 2p^{(k−1)/2}.

9.2.3 An Example of non-semisimple U_{p}

Suppose f = f_{0} is a normalized eigenform. Let W be the space spanned by f_{0}, f_{1} and let V be the space spanned by f_{0}, f_{1}, f_{2}, f_{3}. Then U_{p} acts on V/W by \overline{f}_{2} ↦ 0 and \overline{f}_{3} ↦ \overline{f}_{2}. Thus the matrix of the action of U_{p} on V/W is ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), which is nonzero and nilpotent, hence not semisimple. Since W is invariant under U_{p} this shows that U_{p} is not semisimple on V, i.e., U_{p} is not diagonalizable.

9.3 The Cusp forms are free of rank 1 over T_{C}

9.3.1 Level 1

Suppose N = 1, so Γ_{1}(N) = SL_{2}(Z). Using the Petersson inner product, we see that all the T_{n} are diagonalizable, so S_{k} = S_{k}(Γ_{1}(1)) has a basis

f_{1}, …, f_{d}

of normalized eigenforms where d = dim S_{k}. This basis is canonical up to ordering. Let T_{C} = T ⊗ C be the ring generated over C by all Hecke operators T_{n}. Then, having fixed the basis above, there is a canonical map

T_{C} ↪ C^{d} : T ↦ (λ_{1}, …, λ_{d}),

where f_{i}|T = λ_{i}f_{i}. This map is injective and dim T_{C} = d, so the map is an isomorphism of C-vector spaces.

The form

v = f_{1} + ⋯ + f_{n}

generates S_{k} as a T-module. Note that v is canonical since it does not depend on the ordering of the f_{i}. Since v corresponds to the vector (1, …, 1) and T ≅ C^{d} acts on S_{k} ≅ C^{d} componentwise, this is just the statement that C^{d} is generated by (1, …, 1) as a C^{d}-module.

Recall from Section 4.2 that there is a perfect bilinear pairing S_{k} × T_{C} → C given by

⟨f, T_{n}⟩ = a_{1}(f|T_{n}) = a_{n}(f),

where a_{n}(f) denotes the nth Fourier coefficient of f. Thus we have simultaneously:

1. S_{k} is free of rank 1 over T_{C}, and

2. S_{k} ≅ Hom_{C}(T_{C}, C) as T-modules.

Combining these two facts yields an isomorphism

T_{C} ≅ Hom_{C}(T_{C}, C).  (9.3.1)

This isomorphism sends an element T ∈ T to the homomorphism

X ↦ ⟨v|T, X⟩ = a_{1}(v|T |X).

Since the identification S_{k} ≅ Hom_{C}(T_{C}, C) is canonical and since the vector v is canonical, we see that the isomorphism (9.3.1) is canonical.

Recall that M_{k} has as basis the set of products E_{4}^{a}E_{6}^{b}, where 4a + 6b = k, and S_{k} is the subspace of forms where the constant coefficient of their q-expansion is 0. Thus there is a basis of S_{k} consisting of forms whose q-expansions have coefficients in Q. Let S_{k}(Z) = S_{k} ∩ Z[[q]], be the submodule of S_{k} generated by cusp forms with Fourier coefficients in Z, and note that S_{k}(Z)⊗Q ≅ S_{k}(Q). Also, the explicit formula (∑ a_{n}q^{n})|T_{p} = ∑ a_{np}q^{n} + p^{k−1} ∑ a_{n}q^{np} implies that the Hecke algebra T preserves S_{k}(Z).

Proposition 9.3.1. We have v ∈ S_{k}(Z).

Proof. This is because v = ∑ Tr(T_{n})q^{n}, and, as we observed above, there is a basis so that the matrices T_{n} have integer coefficients, so their traces are integers.

Example 9.3.2. When k = 36, we have

v = 3q + 139656q^{2} − 104875308q^{3} + 34841262144q^{4} + 892652054010q^{5}
− 4786530564384q^{6} + 878422149346056q^{7} + · · · .

The normalized newforms f_{1}, f_{2}, f_{3} are

f_{i} = q + aq^{2} + (−1/72a^{2} + 2697a + 478011548)q^{3} + (a^{2} − 34359738368)q^{4}

(a^{2} − 34359738368)q^{4} + (−69/2a^{2} + 14141780a + 1225308030462)q^{5} + ⋯,

for a each of the three roots of X^{3}−139656X^{2}−59208339456X−1467625047588864.

9.3.2 General level

Now we consider the case for general level N. Recall that there are maps

S_{k}(Γ_{1}(M)) → S_{k}(Γ_{1}(N)),

for all M dividing N and all divisor d of N/M.

Definition 9.3.3. The old subspace of S_{k}(Γ_{1}(N)) is the space generated by all images of these maps with M|N but M ≠ N. The new subspace is the orthogonal complement of the old subspace with respect to the Petersson inner product.

There is an algebraic definition of the new subspace. One defines trace maps

S_{k}(Γ_{1}(N)) → S_{k}(Γ_{1}(M))

for all M < N, M | N which are adjoint to the above maps (with respect to the Petersson inner product). Then f is in the new part of S_{k}(Γ_{1}(N)) if and only if f is in the kernels of all of the trace maps.

It follows from Atkin-Lehner-Li theory that the T_{n} acts semisimply on the new subspace S_{k}(Γ_{1}(M))_{new} for all M ≥ 1, since the common eigenspaces for all T_{n} each have dimension 1. Thus S_{k}(Γ_{1}(M))_{new} has a basis of normalized eigenforms. We have a natural map

⨁_{M|N} S_{k}(Γ_{1}(M))_{new} ↪ S_{k}(Γ_{1}(N)).

The image in S_{k}(Γ_{1}(N)) of an eigenform f for some S_{k}(Γ_{1}(M))_{new} is called a newform of level M_{f} = M. Note that a newform of level less than N is often not an eigenform for all of the Hecke operators acting on S_{k}(Γ_{1}(N)); for example, if N = p^{r}M with p ∤ M a prime, then f is not eigenform for T_{p} (see Equation 9.2.1 above).

Let

v = ∑_{f} f(q^{\frac{N}{M_{f}}} ) ∈ S_{k}(Γ_{1}(N)),

where the sum is taken over all newforms f of weight k and some level M | N. This generalizes the v constructed above when N = 1 and has many of the same good properties. For example, S_{k}(Γ_{1}(N)) is free of rank 1 over T_{C} with basis element v. Moreover, the coefficients of v lie in Z, but to show this we need to know that S_{k}(Γ_{1}(N)) has a basis whose q-expansions lie in Q[[q]]. This is true, but we will not prove it here. One way to proceed is to use the Tate curve to construct a qexpansion map H^{0}(X_{1}(N), Ω_{X_{1}(N)/Q}) → Q[[q]] that is compatible with the usual Fourier expansion map. Example 9.3.4. The space S_{2}(Γ_{1}(22)) has dimension 6. There is a single newform of level 11,

f = q − 2q^{2} − q^{3} + 2q^{4} + q^{5} + 2q^{6} − 2q^{7} + · · · .

There are four newforms of level 22, the four Gal(\overline{Q}/Q)-conjugates of

g = q − ζq^{2} + (−ζ^{3} + ζ − 1)q^{3} + ζ^{2}q^{4} + (2ζ^{3} − 2)q^{5}
+ (ζ^{3} − 2ζ^{2} + 2 ζ − 1)q^{6} − 2ζ^{2}q^{7} + ...

where ζ is a primitive 10th root of unity. Warning 9.3.5. Let S = S_{2}(Γ_{0}(88)), and let v = ∑ Tr(T_{n})q^{n}. Then S has dimension 9, but the Hecke span of v only has dimension 7. Thus the more “canonical looking” element ∑ Tr(T_{n})q^{n} is not a generator for S.

sage: S = CuspForms(88)
sage: B = S.sturm_bound(); B

25

sage: f = QQ[['q']]([0]+[S.T(n).trace() for n in [1..B]], B+1)
sage: f

9*q - 2*q^2 - 4*q^3 - 2*q^5 + 2*q^6 - 8*q^7 + ... + O(q^26)
sage: f = S(f)
sage: span([S.T(n)(f).element() for n in [1..B]]).dimension()

7

Remark 9.3.6. Recall Proposition 4.6.5 that the Fourier coefficients of each normalized eigenform in S_{k} are totally real algebraic integers. A CM field is a quadratic imaginary extension of a totally real field. For example, when n > 2, the field Q(ζ_{n}) is a CM field, with totally real subfield Q(ζ_{n})^{+} = Q(ζ_{n} + 1/ζ_{n}). More generally, one shows that the eigenvalues of any newform f ∈ S_{k}(Γ_{1}(N)) generate a totally real or CM field.

9.4 Decomposing the anemic Hecke algebra

We first observe that it make no difference whether or not we include the Diamond bracket operators in the Hecke algebra. Then we note that the Q-algebra generated by the Hecke operators of index coprime to the level is isomorphic to a product of fields corresponding to the Galois conjugacy classes of newforms. Here the Galois group G_{Q} = Gal(\overline{Q}/Q) acts on newforms by acting on their coefficients, and a Galois conjugacy class means an orbit for the action of G_{Q}.

Proposition 9.4.1. The operators ⟨d⟩ on S_{k}(Γ_{1}(N)) lie in Z[…, T_{n}, …].

Proof. Dirichlet’s theorem on primes in arithmetic progression (see [Lan94, VIII.4]) asserts that every residue class modulo N coprime to N contains infinitely many primes. Thus it is enough to show ⟨p⟩ ∈ Z[…, T_{n}, …] for primes p ∤ N, since each nonzero ⟨d⟩ is of the form ⟨p⟩ for some prime p. Since p ∤ N, we have (see Section 8.2.2) that

T_{p^{2}} = T_{p}^{2} − ⟨p⟩p^{k−1},

so ⟨p⟩p^{k−1} = T_{p}^{2} − T_{p^{2}}. Again, by Dirichlet’s theorem on primes in arithmetic progression, there is another prime q congruent to p mod N. Since p^{k−1} and q^{k−1} are relatively prime, there exist integers a and b such that ap^{k−1} +bq^{k−1} = 1. Then

⟨p⟩ = ⟨p⟩(ap^{k−1} + bq^{k−1}) = a(T_{p}^{2} − T_{p^{2}}) + b(T_{q}^{2} − T_{q^{2}}) ∈ Z[…, T_{n}, …].

Let S be a space of cusp forms, such as S_{k}(Γ_{1}(N)) or S_{k}(Γ_{1}(N), ε). Let

f_{1}, …, f_{d} ∈ S

be representatives for the Galois conjugacy classes of newforms in S of level N_{f_{i}} dividing N. For each i, let K_{i} = Q(…, a_{n}(f_{i}), …) be the field generated by the Fourier coefficients of f_{i}.

Definition 9.4.2 (Anemic Hecke Algebra). The anemic Hecke algebra is the subalgebra

T_{0} = Z[…, T_{n}, …: gcd(n, N) = 1] ⊂ T

of T obtained by adjoining to Z only those Hecke operators T_{n} with n relatively prime to N.

Proposition 9.4.3. We have T_{0} ⊗ Q ≅ ∏_{i=1}^{d} K_{i}, where K_{i} is the number field generated by the eigenvalues of f_{i}.

The map sends T_{n} to (a_{n}(f_{1}), …, a_{n}(f_{d})). The proposition follows from the discussion above and Atkin-Lehner theory. Example 9.4.4. When S = S_{2}(Γ_{1}(22)), then T_{0} ⊗ Q ≅ Q × Q(ζ_{10}) (see Example 9.3.4). When S = S_{2}(Γ_{0}(37)), then T_{0} ⊗ Q ≅ Q × Q. Remark 9.4.5. The index [T : T_{0}] is usually not finite. As explained in Section 9.3.2, the space S_{k}(Γ_{1}(N)) is free of rank 1 over T_{C}, so dim T_{C} = dim S_{k}(Γ_{1}(N)). Proposition 9.4.3 implies that dim(T_{0})_{C} = d is the number of newforms of level dividing N. If there is at least one oldform in S_{k}(Γ_{1}(N)), then d < dim S_{k}(Γ_{1}(N)), because that oldform has at least two linearly independent images in S_{k}(Γ_{1}(N))

80 9. Newforms and Euler Products

Some Explicit Genus Computations

This chapter is about computing the genus of certain modular curves, or equivalently, the dimensions of certain spaces of cusp forms. Section 10.1 explains the connection between genus and the dimension of a space of cusp form, and the general picture regarding ramification of certain covers of modular curves. Then in Section 10.2, we explain our strategy to compute the genus of various modular curves using an Euler characteristic argument. In Section 10.3 we apply this strategy in the case of X(N), and Section 10.4 treats the case of X_{0}(N) with N prime. Finally, in Section 10.5 we discuss a natural isomorphism between two spaces of mod p modular forms, which is suggested by dimension considerations.

We do not treat any other cases explicitly in this book. For the general case of X_{0}(N), the reader might look at [Shi94, §1.6], for X_{1}(N) at [DI95, §9.1], and at [DS05, Ch. 3]. See also Section 5.3 for a different approach in to computing the genus of X(N).

10.1 Computing the dimension of S_{2}(Γ)

Let k = 2 unless otherwise noted, and let Γ ⊂ SL_{2}(Z) be a congruence subgroup. Then S_{2}(Γ) ≅ H^{0}(X_{Γ}, Ω^{1}) where X_{Γ} = (Γ\H) ∪ (Γ\P^{1}(Q)). By definition dim H^{0}(X_{Γ}, Ω^{1}) is the genus of X_{Γ}.

Since Γ ⊂ Γ(1) there is a covering X_{Γ} → X_{Γ(1)} ⟶^{j} P^{1}(C), which can only possibly be ramified at points above 0, 1728, ∞ ∈ P^{1}(C). Here 0 corresponds to ρ = e^{2πi/3} and 1728 corresponds to i under the j-invariant map. We illustrate this as follows:

\xymatrix{Γ\H \ar[r] \ar[d] & X_{Γ} \ar[d] \\ Γ(1)\H \ar[r] & X_{Γ(1)} \ar[d]^{j} \\  & P^{1}(C)}

Example 10.1.1. Suppose Γ = Γ_{0}(N). The degree of the covering is the index (SL_{2}(Z)/{±1} : Γ_{0}(N)/{±1}). As explained in Theorem 5.2.5, a point on Y_{Γ(1)} corresponds to an isomorphism class of elliptic curves over C, whereas a points on Y_{0}(N) correspond to an isomorphism class of a pair consisting of an elliptic curve and a cyclic subgroup of order N. The map Y_{0}(N) → Y_{Γ(1)} = Y_{0}(1) forgets the extra structure of the cyclic subgroup.

10.2 Application of Riemann-Hurwitz

Now we compute the genus of X_{Γ} by applying the Riemann-Hurwitz formula. We recall some standard facts about the Euler characteristic of a topological space, and also the Riemann-Hurwitz formula.

The Euler characteristic χ is additive in the sense that if A and B are disjoint spaces then

χ(A ∪ B) = χ(A) + χ(B).

Also, if X is a compact Riemann surface of genus g, then χ(X) = 2 − 2g, and χ({point}) = 1. Thus

χ(X − {p_{1}, …, p_{n}}) = χ(X) − nχ({point}) = (2 − 2g) − n.

If X → Y is an unramified covering of degree d then a special case of the Riemann-Hurwitz formula is the assertion that

χ(X) = d · χ(Y).  (10.2.1)

Identifying X_{Γ(1)} and P^{1}(C) using the j map, consider the covering

X_{Γ} − {points over 0, 1728, ∞}

X_{Γ(1)} − {0, 1728, ∞}.

This covering is unramified, as was explained in Proposition 5.3.3.

Since X_{Γ(1)} has genus 0, the space X_{Γ(1)} − {0, 1728, ∞} has Euler characteristic 2 − 3 = −1. If we let g = χ(X_{Γ}) then

χ(X_{Γ} − {points over 0, 1728, ∞}) = 2 − 2g − n_{0} − n_{1728} − n_{∞},

where n_{p} denotes the number of points lying over p. Thus −d = 2 − 2g − n_{0} − n_{1728} − n_{∞} whence

2g − 2 = d − n_{0} − n_{1728} − n_{∞}.

Conclusion: Finding the genus of any modular curve X_{Γ} amounts to calculating:

1. the degree d of the j function,

2. the number n_{0} + n_{1728} of points on X_{Γ} with j-invariant 0 or 1728 and,

3. the number n_{∞} of cusps for Γ.

10.3 The Genus of X(N)

Let N > 3 and consider the modular curve X = X(N). There is a natural covering map X → X(1) ⟶^{j} C. Let d be the degree, then as we saw above

2g − 2 = d − n_{0} − n_{1728} − n_{∞}

where g is the genus of X and n_{p} is the number of points lying over p. Since n_{0} = d/3 and n_{1728} = d/2,

2g − 2 = \frac{d}{6} − n_{∞}.

Now we count the number n_{∞} of cusps of X(N), that is, the size of Γ(N)\P^{1}(Q). There is a surjective map from SL_{2}(Z) to P^{1}(Q) given by

(\begin{matrix}a & b \\ c & d \end{matrix}) ↦ (\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}1 \\ 0 \end{matrix}) .

The subgroup of elements of SL_{2}(Z) that stabilize (1, 0) is

U = \{± (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) : a ∈ Z\} .

The cusps of X(N) are the elements of

Γ(N)\(SL_{2}(Z)/U) = (Γ(N)\ SL_{2}(Z))/U = SL_{2}(Z/NZ)/U

which has order

\frac{# SL_{2}(Z/NZ)}{2N} = \frac{d}{N} .

Substituting this into the above formula gives

2g − 2 = \frac{d}{6} − \frac{d}{N} = \frac{d}{6N} (N − 6),

so

g = 1 + \frac{d}{12N} (N − 6).

When N is prime,

d = \frac{1}{2} · # SL_{2}(Z/NZ) = \frac{1}{2} · \frac{(N^{2} − 1)(N^{2} − N)}{N − 1}.

Thus when N = 5, d = 60 so g = 0, and when N = 7, d = 168 so g = 3. When N = 2011, we have g = 337852526.

10.4 The Genus of X_{0}(N), for N prime

Suppose N > 3 and N is prime. The covering map X_{0}(N) → X(1) is of degree N +1 since a point of X_{0}(N) corresponds to an elliptic curve along with a subgroup of order N and there are N + 1 such subgroups because N is prime. Since N is prime, X_{0}(N) has two cusps; they are the orbit of ∞ which is unramified and 0 which is ramified of order N. Thus

2g − 2 = N + 1 − 2 − n_{1728} − n_{0}.

Here n_{0} is the number of pairs (E, C) (modulo isomorphism) such that E has j-invariant 0. So we consider E = C/Z[\frac{−1+i\sqrt{3}}{2}] which has endomorphism ring End(E) = Z[μ_{6}]. Now μ_{6}/±1 acts on the cyclic subgroups C so, letting ω be a primitive cube root of unity, we have

(E, C) ≅ (E, ωC) ≅ (E, ω^{2}C).

This might lead one to think that n_{0} is (N + 1)/3, but it may be bigger if, for example, C = ωC. Thus we must count those C so that ωC = C or ω^{2}C = C, that is, those C which are stable under O = Z[\frac{−1+i\sqrt{3}}{2}]. So we must compute the number of stable O/NO-submodules of order N. This depends on the structure of O/NO:

O/NO = \begin{cases}F_{N} ⊕ F_{N} & if (\frac{−3}{N}) = 1 (N splits) \\ F_{N^{2}} & if (\frac{−3}{N}) = −1 (N stays inert) \end{cases}

Since O/NO = F_{N^{2}} is a field it has no submodules of order N, whereas F_{N} ⊕ F_{N} has two O/NO-submodules of order N, namely F_{N} ⊕ 0 and 0 ⊕ F_{N}. Thus

n_{0} = \begin{cases}\frac{N+1}{3} & if N ≡ 2 (mod 3) \\ \frac{N−1}{3} + 2 & if N ≡ 1 (mod 3) \end{cases}

Exercise 10.4.1. It is an exercise in elegance to write this as a single formula involving the quadratic symbol.

By similar reasoning one shows that

n_{1728} = \begin{cases}\frac{N+1}{2} & if N ≡ 3 (mod 4) \\ \frac{N−1}{2} + 2 & if N ≡ 1 (mod 4) \end{cases}

We can now compute the genus of X_{0}(N) for any prime N. For example, if N = 37 then 2g − 2 = 36 − (2 + 18) − (14) = 2 so g = 2. Similarly, X_{0}(13) has genus 0 and X_{0}(11) has genus 1. In general, X_{0}(N) has genus approximately N/12.

Serre constructed a nice formula for the above genus. Suppose N > 3 is a prime and write N = 12a + b with 0 ≤ b ≤ 11. Then Serre’s formula is

\begin{matrix}b & 1 & 5 & 7 & 11 \\ g & a − 1 & a & a & a + 1 \end{matrix}

10.5 Modular forms mod p

Let N be a positive integer, let p be a prime and assume Γ is either Γ_{0}(N) or Γ_{1}(N).

10.5 Modular forms mod p 85

Definition 10.5.1. Let M_{k}(Γ, Z) = M_{k}(Γ, C) ∩ Z[[q]]. Then

M_{k}(Γ, F_{p}) = M_{k}(Γ, Z) ⊗_{Z} F_{p}

is the space of modular forms mod p of weight k.

Suppose p = N. Then one has Serre’s Equality:

M_{p+1}(SL_{2}(Z), F_{p}) = M_{2}(Γ_{0}(p), F_{p})

[[TODO: Insert a dimension formula calculation to give evidence for this equality, since that’s the whole reason this section is in this chapter.]]

The map from the right hand side to the left hand side is accomplished via a certain normalized Eisenstein series. Recall from Section 4.1 that for SL_{2}(Z) we have Eisenstein series

G_{k} = −\frac{B_{k}}{2k} + ∑_{n=1}^{∞} (∑_{d|n} d^{k−1}) q^{n}

and

E_{k} = 1 − \frac{2k}{B_{k}} ∑_{n=1}^{∞}(∑_{d|n} d^{k−1})q^{n}.

One finds ord_{p}(−\frac{B_{k}}{2k}) using Kummer congruences [[TODO: elaborate]]. In particular, ord_{p}(B_{p−1}) = −1, so E_{p−1} ≡ 1 (mod p). Thus multiplication by E_{p−1} increases the weight by p − 1 but does not change the q-expansion mod p. We thus get a map

M_{2}(Γ_{0}(p), F_{p}) → M_{p+1}(Γ_{0}(p), F_{p}).

The map

M_{p+1}(Γ_{0}(p), F_{p}) → M_{p+1}(SL_{2}(Z), F_{p})

is the trace map (which is dual to the natural inclusion going the other way) and is accomplished by averaging in order to get a form invariant under SL_{2}(Z).

86 10. Some Explicit Genus Computations

The Field of Moduli

In this chapter we will study the field of definition of the modular curve X(N). We assume the reader if familiar with the correspondence between function fields and nonsingular projective algebraic curves, as explained in [Har77, §I.6]. For example, the function field of P_{Q}^{1} is Q(t). We will also freely use basic facts about the jinvariant of an elliptic curve, including that two curves are isomorphic over an algebraically closed field if and only if they have the same j-invariant.

11.1 Algebraic definition of X(N)

If E is an elliptic curve given by a Weierstrass equation y^{2} = 4x^{3} − g_{2}x − g_{3}, then

j(E) = j(g_{2}, g_{3}) = \frac{1728g_{2}^{3}}{g_{2}^{3} − 27g_{3}^{2}} .

The j-invariant determines the isomorphism class of E over C, since j : X(1) → P_{C}^{1} has degree 1 (see [Ser73, VII.3.3]). There is an elliptic curve E/Q(t) such that j(E) = t, for example, the elliptic curve with Weierstrass equation

y^{2} = 4x^{3} − \frac{27t}{t − 1728}x − \frac{27t}{t − 1728}.  (11.1.1)

For the moment, let k be any field, E/k an arbitrary elliptic curve and N a positive integer prime to char k. Upon fixing a choice of basis for E[N], we have E[N](\overline{k}) ≅ (Z/NZ)^{2}. Let k(E[N]) be the field obtained by adjoining the coordinates of the N-torsion points of E to k, so we have a tower of fields \overline{k} ⊃ k(E[N]) ⊃ k. There is a Galois representation on the N torsion of E:

Gal(\overline{k}/k) ⟶^{ρ_{E}} Aut(E[N]) ≅ GL_{2}(Z/NZ)

and Gal(\overline{k}/k(E[N])) = ker(ρ_{E}). Also Gal(k(E[N])/k) ↪ GL_{2}(Z/NZ).

Applying the above observations with k = Q(t) and E the curve from (11.1.1) with j-invariant t, shows that the Galois group of the extension Q(t)(E[N]) of Q(t) is contained in GL_{2}(Z/NZ). Let X(N) be the algebraic curve corresponding to the function field Q(t)(E[N]). As we will see in Proposition 11.4.2 below,

\overline{Q} ∩ (Q(t)(E[N])) ⊂ Q(μ_{N}),

so the curve X(N) is defined over Q(μ_{N}).

The main input we need is that the representation ρ_{E} is surjective. Our strategy to prove this starts by composing ρ_{E} with the natural map GL_{2}(Z/NZ) → GL_{2}(Z/NZ)/{±1} to define a map

\overline{ρ}_{E} : Gal(\overline{k}/k) → GL_{2}(Z/NZ)/\{±1\}.

The following proposition shows that we loose little in passing to \overline{ρ}_{E}.

Proposition 11.1.1. \overline{ρ}_{E} is surjective if and only if ρ_{E} is surjective.

Proof. If \overline{ρ}_{E} is surjective then either ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) or its negative lies in the image of ρ. Thus ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) lies in the image of ρ_{E}. Since \overline{ρ}_{E} is surjective this implies that ρ_{E} is surjective. The converse is trivial.

11.2 Digression on moduli

Recall from Theorem 5.2.5 that the non-cuspidal points in X_{0}(N)(C) are the set of C-isomorphism classes of pairs (E, C) where E/C is an elliptic curve and C is a cyclic subgroup of order N. Assume for the moment that X_{0}(N) has a some sort of canonical model of algebraic curve over Q, and let Y_{0}(N) = X_{0}(N) − { cusps }. It is reasonable to assume that Y_{0}(N)(\overline{Q}) is the set of \overline{Q}-isomorphism classes of pairs (E, C) where E/\overline{Q} is an elliptic curve and C is a cyclic subgroup of order N. Each σ ∈ Gal(\overline{Q}/Q) would then act on the set of these points by sending the class of (E, C) to the class of (^{σ}E,^{σ}C), where ^{σ}E is the elliptic curve got by applying σ to the coefficients of an equation for E, and ^{σ}C ⊂ ^{σ}E(\overline{Q})[N] is the image of C under σ.

Let K be a number field. Since \overline{Q} is a separable extension of K, we have

X_{0}(N)(K) = X_{0}(N)(\overline{Q})^{Gal(\overline{Q}/K)}.

Thus Y_{0}(N)(K) is the set of isomorphism classes of pairs (E, C) ∈ Y_{0}(N)(\overline{Q}) such that for all σ ∈ Gal(\overline{Q}/K), there exists some isomorphism (^{σ}E,^{σ}C) ≈ (E, C) defined over \overline{Q}. There is a map

{K-isomorphism classes of pairs (E, C)/K} → Y_{0}(N)(K)

that is “notoriously” non-injective. For example, when N = 1, the map X_{0}(1) ⟶^{j} P^{1} identifies X_{0}(1) with the j-line. With this identification, the above map then sends an elliptic curve E/K to its j-invariant. Since all quadratic twists of E are also defined over K and have the same j-invariant, there are infinitely many different elements of the left hand side that all map to the same point j = j(E) in the right hand side.

11.3 When is ρ_{E} surjective? 89

The paper [DR73] contains a proof that the map is surjective, i.e., that any isomorphism class [(E, C)], with E and C defined over \overline{Q} that is fixed by all σ ∈ Gal(\overline{Q}/K) has a representative element (E, C) with E and C defined over K. When N = 1, [DR73] observe that the map is surjective directly, i.e., they answer this question:

Question 11.2.1. If E/\overline{K} is isomorphic to all its Galois conjugates, is there a curve E′/K that is isomorphic to E over \overline{K}?

For N > 1 they show that certain obstructions vanish. [[TODO: precise ref?]]

11.3 When is ρ_{E} surjective?

Let K be a field of characteristic 0.

Proposition 11.3.1. Let E_{1} and E_{2} be elliptic curves defined over K with equal j-invariants, so E_{1} ≈ E_{2} over \overline{K}. Assume E_{1} and E_{2} do not have complex multiplication over \overline{K}. Then ρ_{E_{1}} is surjective if and only if ρ_{E_{2}} is surjective.

Proof. Assume ρ_{E_{1}} is surjective. Since E_{1} does not have complex multiplication over \overline{K}, we have Aut E_{1} = \{±1\}. Choose an isomorphism φ : E_{1} ⟶^{∼} E_{2} over \overline{K}. Then for any σ ∈ Gal(\overline{K}/K) we have the diagram

\xymatrix{E_{1} \ar[r]^{φ} \ar[d]_{=} & E_{2} \ar[d]_{=} \\ ^{σ}E_{1} \ar[r]^{σφ} & ^{σ}E_{2}},

where we note that ^{σ}E_{1} = E_{1} and ^{σ}E_{1} = E_{2}, since both curves are defined over K. Thus σφ = ±φ for all σ ∈ Gal(\overline{K}/K), so φ : E_{1}[N] → E_{2}[N] defines an equivalence \overline{ρ}_{E_{1}} ≅ \overline{ρ}_{E_{2}}. Since ρ_{E_{1}} is surjective this implies that \overline{ρ}_{E_{2}} is surjective; Proposition 11.1.1 then implies that ρ_{E_{2}} is surjective.

Let K = C(j), with j transcendental over C. Let E/K be an elliptic curve such as (11.1.1) with j-invariant j. Fix a positive integer N and let

ρ_{E} : Gal(\overline{K}/K) → GL_{2}(Z/NZ)

be the associated Galois representation on the N-torsion of E. Then one can prove using an algebraic definition of the Weil pairing that det ρ_{E} is the cyclotomic character, which is trivial since C ⊂ K and C contains the Nth roots of unity. Thus the image of ρ_{E} lands inside of SL_{2}(Z/NZ). Our next theorem states that a “generic elliptic curve”, i.e., a curve with j-invariant j, has maximal possible Galois action on its division points.

Theorem 11.3.2. ρ_{E} : Gal(\overline{K}/K) → SL_{2}(Z/NZ) is surjective.

Igusa [Igu56] [[TODO: I could not find anything about this in [Igu56], but I only looked for a few minutes. In [DR73, §2] they say that Igusa proves this, though, and sketch his

proof.]] found an algebraic proof of this theorem, but we content ourselves with making some comments on how an analytic proof goes.

Proof. The field C(j) = K = F_{1} is the field of modular functions for SL_{2}(Z). Suppose N ≥ 3 and let F_{N} be the field of mereomorphic functions for Γ(N), i.e., meromorphic functions on 𝔥^{∗} that are invariant under Γ(N). One can prove [] that the extension F_{N}/F_{1} is a Galois extension with Galois group SL_{2}(Z/NZ)/{±1} ≅ \overline{Γ(1)}/\overline{Γ(N)}, where \overline{Γ(N)} denotes the image of Γ(N) in PSL_{2}(Z).

Let E be an elliptic curve over K with j-invariant j. We will show that Gal(F_{N}/F_{1}) acts transitively on the x-coordinates of the N-torsion points of E. This will show that \overline{ρ}_{E} maps surjectively onto SL_{2}(Z/NZ)/\{±1\} [[TODO: Why?]]. Then by Proposition 11.1.1, ρ_{E} maps surjectively onto SL_{2}(Z/NZ), as claimed. [[TODO: Many

details need to be added based on Rohrlich’s proof!]]

We will now view the x-coordinates of the points in E[N] as functions on H that are invariant under Γ(N). (Thus K(E[N]/{±1}) ⊂ F_{N}.) Let τ ∈ H and let Λ_{τ} = Zτ + Z. Consider ℘(z, Λ_{τ}), which gives the x-coordinate of C/Λ_{τ} in its standard form y^{2} = 4x^{3}−g_{2}x−g_{3}. Define, for each nonzero (r, s) ∈ (Z/NZ)^{2}/{±1}, a function

f_{(r,s)} : H → C : τ ↦ \frac{g_{2}(τ)}{g_{3}(τ)}℘ (\frac{rτ + s}{N}, Λ_{τ} ) .

We first prove that for any α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z),

f_{(r,s)}(ατ) = f_{(r,s)( \begin{matrix}a & b \\ c & d \end{matrix} )}(τ),

as follows. Indeed, ℘ is homogeneous of degree −2, g_{2} is modular of weight 4 and

g_{3} is modular of weight 6, so [[TODO: Weierstrass ℘ notation below is missing second argument.]]

f_{(r,s)}(ατ) = \frac{g_{2}(ατ)}{g_{3}(ατ)}℘ (\frac{rατ + s}{N})

= (cτ + d)^{−2} \frac{g_{2}(τ)}{g_{3}(τ)}℘ (\frac{raτ + rb + csτ + sd}{N(cτ + d)})

= \frac{g_{2}(τ)}{g_{3}(τ)}℘ (\frac{(ra + sc)τ + rb + sd}{N}) = f_{(r,s)α}(τ)

Let E_{j(τ)} denote an elliptic curve over C with j-invariant j(τ). If τ ∈ H with g_{2}(τ), g_{3}(τ) ≠ 0 then the f_{(r,s)}(τ) are the x-coordinates of the nonzero N-division points of E_{j(τ)}. The various f_{(r,s)}(τ) are distinct. Thus SL_{2}(Z/NZ)/{±1} acts transitively on the f_{(r,s)}. The consequence is that the N^{2}−1 nonzero division points of our generic curve E have x-coordinates in \overline{F}_{N} equal to the f_{(r,s)} ∈ F_{N}.

11.4 Observations

Proposition 11.4.1. If E/Q(μ_{N})(t) is an elliptic curve with j(E) = t, then ρ_{E} has image SL_{2}(Z/NZ).

Proof. Since Q(μ_{N}) contains the Nth roots of unity, the Nth cyclotomic character is trivial, hence the determinant of ρ_{E} is trivial. Thus the image of ρ_{E} lies in SL_{2}(Z/NZ). In the other direction, there is a natural inclusion

SL_{2}(Z/NZ) = Gal(C(t)(E[N])/C(t)) ↪ Gal(Q(μ_{N})(t)(E[N])/Q(μ_{N})(t)).

Proposition 11.4.2. If E/Q(t) is an elliptic curve with j(E) = t, then ρ_{E} has image GL_{2}(Z/NZ) and \overline{Q} ∩ (Q(t)(E[N])) = Q(μ_{N}).

Proof. Since Q(t) contains no Nth roots of unity (recall N ≥ 3), the mod N cyclotomic character, and hence det ρ_{E}, is surjective onto (Z/NZ)^{∗}. Since the image of ρ_{E} already contains SL_{2}(Z/NZ) it must equal GL_{2}(Z/NZ). For the second assertion consider the diagram

\overline{Q}  Q(t)(E[N])_{SL_{2}} Q(μ_{N})_{(Z/NZ)^{∗}}  Q(μ_{N})(t)_{GL_{2} / SL_{2}=(Z/NZ)^{∗}} Q  Q(t)

This gives a way to view X_{0}(N) as a projective algebraic curve over Q. Let K = Q(t) and let L = K(E[N]) ⊃ Q(μ_{N})(t). Then

H = \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )\} ⊂ GL_{2}(Z/NZ) = Gal(L/K).

The fixed field L^{H} is an extension of Q(t) of transcendence degree 1 with field of constants \overline{Q} ∩ L^{H} = Q, i.e., a projective algebraic curve defined over Q.

11.5 A descent problem

Consider the following exercise, which may be approached in an honest or dishonest way. Exercise 11.5.1. Suppose L/K is a finite Galois extension and G = Gal(L/K). Let E/L be an elliptic curve, assume Aut_{L} E = {±1}, and suppose that for all g ∈ G, there is an isomorphism ^{g}E ⟶^{∼} E over L. Show that there exists E_{0}/K such that E_{0} ≅ E over L.

Caution! The exercise is false as stated. Both the dishonest and honest approaches below work only if L is a separable closure of K. Now: can one construct a counterexample?

Discussion. First the hard, but “honest” way to look at this problem. For notions on descent see [Ser88, V.20]. By descent theory, to give E_{0} is the same as to give a family (λ_{g})_{g∈G} of maps λ_{g} : ^{g}E ⟶^{∼} E such that λ_{gh} = λ_{g} ∘ ^{g}λ_{h} where gλ_{h} = g ∘ λ_{h} ∘ g^{−1}. Note that λ_{g} ∘^{g}λ_{h} maps ^{gh}E → E. This is the natural condition to impose, because if f : E_{0} ⟶^{∼} E and we let λ_{g} = f ∘^{g}(f^{−1}) then λ_{gh} = λ_{g} ∘^{g}λ_{h}.

Using our hypothesis choose, for each g ∈ G, an isomorphism

λ_{g} : ^{g}E ⟶^{∼} E.

Define a map c by

c(g, h) = λ_{g} ∘^{g}λ_{h} ∘ λ_{gh}^{−1}.

Note that c(g, h) ∈ Aut E = {±1} so c defines an element of

H^{2}(G, \{±1\}) ⊂ H^{2}(Gal(\overline{L}/K), \{±1\}) = Br(K)[2].

Here Br(K)[2] denotes the 2-torsion of the Brauer group

Br(K) = H^{2}(Gal(\overline{L}/K), \overline{L}^{∗}).

This probably leads to an honest proof.

The dishonest approach is to note that g(j(E)) = j(E) for all g ∈ G, since all conjugates of E are isomorphic and j(^{g}E) = g(j(E)). Thus j(E) ∈ K (assuming K is perfect), so we can define E_{0}/K by substituting j(E) into (11.1.1). This gives an elliptic curve E_{0} defined over K but isomorphic to E over \overline{K}.

11.6 Second look at the descent exercise

We have been discussing the following problem. Suppose L/K is a Galois extension with char K = 0, and let E/L be an elliptic curve. Suppose that for all σ ∈ G = Gal(L/K), ^{σ}E ≅ E over L. Conclude that there is an elliptic curve E_{0}/K such that E_{0} ≅ E over L. The conclusion may fail to hold if L is a finite extension of K, but the exercise is true when L = \overline{K}. First we give a descent argument which holds when L = \overline{K} and then give a counterexample to the more general statement.

For g, h ∈ G = Gal(L/K) we define an automorphism c(g, h) ∈ Aut E = {±1}. Choose for every g ∈ Gal(L/K) some isomorphism

λ_{g} : ^{g}E ⟶^{∼} E.

If the λ_{g} were to all satisfy the compatibility criterion λ_{gh} = λ_{g} ∘ ^{g}λ_{h} then by descent theory we could find a K-structure on E, that is a model for E defined over K and isomorphic to E over L. Define c(g, h) by c(g, h)λ_{gh} = λ_{g} ∘^{g}λ_{h} so c(g, h) measures how much the λ_{g} fail to satisfy the compatibility criterion. Since c(g, h) is a cocycle it defines an elements of H^{2}(G, {±1}). We want to know that this element is trivial. When L = \overline{K}, the map H^{2}(G, \{±1\}) → H^{2}(G, L^{∗}) is injective. To see this first consider the exact sequence

0 → \{±1\} → \overline{K}^{∗} ⟶^{2} \overline{K}^{∗} → 0

where 2 : \overline{K}^{∗} → \overline{K}^{∗} is the squaring map. Taking cohomology yields an exact sequence

H^{1}(G, \overline{K}^{∗}) → H^{2}(G, \{±1\}) → H^{2}(G, \overline{K}^{∗}).

By Hilbert’s theorem 90 ([Ser79] Ch. X, Prop. 2), H^{1}(G, {±1}) = 0. Thus we have an exact sequence

0 → H^{2}(G, \{±1\}) → H^{2}(G, \overline{K}^{∗})[2] → 0.

Thus H^{2}(G, {±1}) naturally sits inside H^{2}(G, L^{∗}).

[[TODO: To finish Ribet does something with differentials and H^{0}(^{g}E, Ω^{1}) which I don’t understand.]]

The counterexample in the case when L/K is finite was provided by Kevin Buzzard [[TODO: (who said Coates gave it to him)]]. Let L = Q(i), K = Q and E be

11.7 Action of GL_{2} 93

the elliptic curve with Weirstrass equation iy^{2} = x^{3} + x + 1. Then E is isomorphic to its conjugate over L but one can show directly that E has no model over Q.

[[TODO: BUT: Isn’t y^{2} = x^{3} + x + 1 a model for this curve over Q. It is isomorphic to that curve over \overline{Q}.]]

11.7 Action of GL_{2}

Let N > 3 be an integer and E/Q(j) an elliptic curve with j-invariant j(E) = j. Then there is a Galois extension

\begin{matrix}F_{N} = Q(j)(E[N]/\{±1\}) \\ F_{1} = Q(j) \end{matrix}

with Galois group GL_{2}(Z/NZ)/{±1}. Think of Q(j)(E[N]/{±1}) as the field obtained from Q(j) by adjoining the x-coordinates of the N-torsion points of E. Note that this situation differs from the previous situation in Section 11.3 in that the base field C has been replaced by Q.

Consider

F = ⋃_{N} F_{N}

which corresponds to a projective system of modular curves . Let A_{f} be the ring of finite adèles; thus

A_{f} = Q̂ = Ẑ ⊗ Q ⊂ ∏_{p} Q_{p}.

We think of A_{f} as

{(x_{p}) ∈ ∏ Q_{p} : x_{p} ∈ Z_{p} for almost all p}.

The group GL_{2}(A_{f}) acts on F. To understand what this action is we first consider the subgroup GL_{2}(Ẑ) of GL_{2}(A_{f}).

It can be shown that

F = Q(f_{N,(r,s)} : (r, s) ∈ (Z/NZ)^{2} − {(0, 0)}, N ≥ 1)

where f_{N,(r,s)} is as defined in Section 11.3. We define an action of GL_{2}(Ẑ) on F as follows. To describe how g ∈ GL_{2}(Ẑ) acts on f_{N,(r,s)}, first map g into GL_{2}(Z/NZ) via the natural reduction map, then note that GL_{2}(Z/NZ) acts on f_{N,(r,s)} by

(\begin{matrix}a & b \\ c & d \end{matrix}) · f_{N,(r,s)} = f_{N,(r,s)( \begin{matrix}a & b \\ c & d \end{matrix} )} = f_{N,(ra+sc,rb+sd)}.

Let E be an elliptic curve. Then the universal Tate module is

T (E) = \underleftarrow{lim}_{N≥1} E[N] = ∏_{p} T_{p}(E).

There is some isomorphism α : Ẑ^{2} ⟶^{∼} T (E). Via right composition, GL_{2}(Ẑ) acts on the collection of all such isomorphisms α. So GL_{2}(Ẑ) acts naturally on pairs (E, α) with the action doing nothing to E. An important point to be grasped when constructing objects like Shimura varieties is that we must “free ourselves” and allow GL_{2}(Ẑ) to act on the E’s as well.

Let

g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}^{+}(Q).

Let τ ∈ H and let E = E_{τ} be the elliptic curve determined by the lattice Λ_{τ} = Zτ + Z. Let

α_{τ} : Λ_{τ} = Zτ + Z ⟶^{∼} Z^{2}

be the isomorphism defined by τ ↦ (1, 0) and 1 ↦ (0, 1). Now view α = α_{τ} as a map

α : Z^{2} ⟶^{∼} H_{1}(E(C), Z).

Tensoring with Q then gives another map (also denoted α)

α : Q^{2} ⟶^{∼} H_{1}(E, Q).

Then α ∘ g is another isomorphism

Q^{2} ⟶^{α∘g} H_{1}(E, Q),

which induces an isomorphism Z^{2} ⟶^{∼} L′ ⊂ H_{1}(E, Q) where L′ is a lattice. There exists an elliptic curve E′/C and a map λ ∈ Hom(E′, E) ⊗ Q, which induces a map (also denoted λ)

λ : H_{1}(E′, Z) ⟶^{∼} L′ ⊂ H_{1}(E, Q)

on homology groups.

Now we can define an action on pairs (E, α) by sending (E, α) to (E′, α′). Here α′ is the map α′ : Z^{2} → H_{1}(E′, Z) given by the composition

Z^{2} ⟶^{αg} L′ ⟶^{λ^{−1}} H_{1}(E′, Z).

In more concrete terms the action is

g : (E_{τ} , α_{τ}) ↦ (E′_{τ} , α′_{τ})

where τ′ = gτ = \frac{aτ+b}{cτ+d}.

Hecke Operators as Correspondences

Our goal is to view the Hecke operators T_{n} and ⟨d⟩ as objects defined over Q that act in a compatible way on modular forms, modular Jacobians, and homology. In order to do this, we will define the Hecke operators as correspondences.

12.1 The Definition

Definition 12.1.1 (Correspondence). Let C_{1} and C_{2} be curves. A correspondence C_{1} ⇝ C_{2} is a curve C together with nonconstant morphisms α : C → C_{1} and β : C → C_{2}. We represent a correspondence by a diagram

\xymatrix{ & C \ar[dl]_{α} \ar[dr]^{β} \\ C_{1} &  & C_{2}}

Given a correspondence C_{1} ⇝ C_{2} the dual correspondence C_{2} ⇝ C_{1} is obtained by looking at the diagram in a mirror

\xymatrix{ & C \ar[dl]_{β} \ar[dr]^{α} \\ C_{2} &  & C_{1}}

In defining Hecke operators, we will focus on the simple case when the modular curve is X_{0}(N) and Hecke operator is T_{p}, where p ∤ N. We will view T_{p} as a correspondence X_{0}(N) ⇝ X_{0}(N), so there is a curve C = X_{0}(pN) and maps α and β fitting into a diagram

\xymatrix{ & X_{0}(pN) \ar[dl]_{α} \ar[dr]^{β} \\ X_{0}(N) &  & X_{0}(N).}

The maps α and β are degeneracy maps which forget data. To define them, we view X_{0}(N) as classifying isomorphism classes of pairs (E, C), where E is an elliptic curve and C is a cyclic subgroup of order N (we will not worry about what happens at the cusps, since any rational map of nonsingular curves extends uniquely to a morphism [Har77, Ch. I, Prop. 6.8]). Similarly, X_{0}(pN) classifies isomorphism classes of pairs (E, G) where G = C ⊕ D, C is cyclic of order N and D is cyclic of order p. Note that since (p, N) = 1, the group G is cyclic of order pN and the subgroups C and D are uniquely determined by G. The map α forgets the subgroup D of order p, and β quotients out by D:

α : (E, G) ↦ (E, C)  (12.1.1)
β : (E, G) ↦ (E/D, (C + D)/D)  (12.1.2)

We translate this into the language of complex analysis by thinking of X_{0}(N) and X_{0}(pN) as quotients of the upper half plane. The first map α corresponds to the map

Γ_{0}(pN)\𝔥 → Γ_{0}(N)\𝔥

induced by the inclusion Γ_{0}(pN) ↪ Γ_{0}(N). The second map β is constructed by composing the isomorphism

Γ_{0}(pN)\𝔥 ⟶^{∼} (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) Γ_{0}(pN) (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})^{−1} \𝔥  (12.1.3)

with the map to Γ_{0}(N)\𝔥 induced by the inclusion

(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) Γ_{0}(pN) (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})^{−1} ⊂ Γ_{0}(N).

The isomorphism (12.1.3) is induced by z ↦ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) z = pz; explicitly, it is

Γ_{0}(pN)z ↦ ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) Γ_{0}(pN) ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )^{−1} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) z.

(Note that this is well-defined.)

The maps α and β induce pullback maps on differentials

α^{∗}, β^{∗} : H^{0}(X_{0}(N), Ω^{1}) → H^{0}(X_{0}(pN), Ω^{1}).

We can identify S_{2}(Γ_{0}(N)) with H^{0}(X_{0}(N), Ω^{1}) by sending the cusp form f(z) to the holomorphic differential f(z)dz. Doing so, we obtain two maps

α^{∗}, β^{∗} : S_{2}(Γ_{0}(N)) → S_{2}(Γ_{0}(pN)).

Since α is induced by the identity map on the upper half plane, we have α^{∗}(f) = f, where we view f = ∑ a_{n}q^{n} as a cusp form with respect to the smaller group Γ_{0}(pN). Also, since β^{∗} is induced by z ↦ pz, we have

β^{∗}(f) = p ∑_{n=1}^{∞} a_{n}q^{pn}.

The factor of p is because

β^{∗}(f(z)dz) = f(pz)d(pz) = pf(pz)dz.

Let X, Y, and C be curves, and α and β be nonconstant holomorphic maps, so we have a correspondence

\xymatrix{ & C \ar[dl]_{α} \ar[dr]^{β} \\ X &  & Y.}

By first pulling back, then pushing forward, we obtain induced maps on differentials

H^{0}(X, Ω^{1}) ⟶^{α^{∗}} H^{0}(C, Ω^{1}) ⟶^{β_{∗}} H^{0}(Y, Ω^{1}).

The composition β_{∗} ∘ α^{∗} is a map H^{0}(X, Ω^{1}) → H^{0}(Y, Ω^{1}). If we consider the dual correspondence, which is obtained by switching the roles of X and Y, we obtain a map H^{0}(Y, Ω^{1}) → H^{0}(X, Ω^{1}).

Now let α and β be as in (12.1.1). Then we can recover the action of T_{p} on modular forms of weight 2 by considering the induced map

β_{∗} ∘ α^{∗} : H^{0}(X_{0}(N), Ω^{1}) → H^{0}(X_{0}(N), Ω^{1})

and using that S_{2}(Γ_{0}(N)) ≅ H^{0}(X_{0}(N), Ω^{1}).

12.2 Maps induced by correspondences

In this section we will see how correspondences induce maps on divisor groups, which in turn induce maps on Jacobians.

Suppose φ : X → Y is a morphism of curves. Let Γ ⊂ X × Y be the graph of φ. This gives a correspondence

\xymatrix{ & Γ \ar[dl]_{α} \ar[dr]^{β} \\ X &  & Y}

We can reconstruct φ from the correspondence by using that φ(x) = β(α^{−1}(x)).

More generally, suppose Γ is a curve and that α : Γ → X has degree d ≥ 1. View α^{−1}(x) as a divisor on Γ (it is the formal sum of the points lying over x, counted with appropriate multiplicities). Then β(α^{−1}(x)) is a divisor on Y. We thus obtain a map

Div^{n}(X) ⟶^{β∘α^{−1}} Div^{dn}(Y),

where Div^{n}(X) is the group of divisors of degree n on X. In particular, setting d = 0, we obtain a map Div^{0}(X) → Div^{0}(Y).

We now apply the above construction to T_{p}. Recall that T_{p} is the correspondence

\xymatrix{ & X_{0}(pN) \ar[dl]_{α} \ar[dr]^{β} \\ X_{0}(N) &  & X_{0}(N),}

where α and β are as in Section 12.1 and the induced map is

(E, C) ↦^{α^{∗}} ∑_{D∈E[p]}(E, C ⊕ D) ↦^{β_{∗}} ∑_{D∈E[p]}(E/D, (C + D)/D).

Thus we have a map Div(X_{0}(N)) → Div(X_{0}(N)). This strongly resembles the first definition we gave of T_{p} on level 1 forms, where T_{p} was a correspondence of lattices.

12.3 Induced maps on Jacobians of curves

Let X be a curve of genus g over a field k. Recall that there is an important association

{ curves X/k } ⟶ { Jacobians Jac(X) = J(X) of curves }

between curves and their Jacobians.

Definition 12.3.1 (Jacobian). Let X be a curve of genus g over a field k. Then the Jacobian of X is an abelian variety of dimension g over k whose underlying group is functorially isomorphic to the group of divisors of degree 0 on X modulo linear equivalence. (For a more precise definition, see Section ?? (Jacobians section)^{1} .) 1

There are many constructions of the Jacobian of a curve. We first consider the Albanese construction. Recall that over C, any abelian variety is isomorphic to C^{g}/L, where L is a lattice (and hence a free Z-module of rank 2g). There is an embedding

ι : H_{1}(X, Z) ↪ H^{0}(X, Ω^{1})^{∗}
γ ↦ ∫_{γ} •

Then we realize Jac(X) as a quotient

Jac(X) = H^{0}(X, Ω^{1})^{∗}/ι(H_{1}(X, Z)).

In this construction, Jac(X) is most naturally viewed as covariantly associated to X, in the sense that if X → Y is a morphism of curves, then the resulting map H^{0}(X, Ω^{1})^{∗} → H^{0}(Y, Ω^{1})^{∗} on tangent spaces induces a map Jac(X) → Jac(Y).

There are other constructions in which Jac(X) is contravariantly associated to X. For example, if we view Jac(X) as Pic^{0}(X), and X → Y is a morphism, then pullback of divisor classes induces a map Jac(Y) = Pic^{0}(Y) → Pic^{0}(X) = Jac(X).

If F : X ⇝ Y is a correspondence, then F induces an a map Jac(X) → Jac(Y) and also a map Jac(Y) → Jac(X). If X = Y, so that X and Y are the same, it can often be confusing to decide which duality to use. Fortunately, for T_{p}, with p prime to N, it does not matter which choice we make. But it matters a lot if p | N since then we have non-commuting confusable operators and this has resulted in mistakes in the literature.

12.4 More on Hecke operators

Our goal is to move things down to Q from C or \overline{Q}. In doing this we want to understand T_{n} (or T_{p}), that is, how they act on the associated Jacobians and how they can be viewed as correspondences. In characteristic p the formulas of Eichler-Shimura will play an important role.

1insert this

We consider T_{p} as a correspondence on X_{1}(N) or X_{0}(N). To avoid confusion we will mainly consider T_{p} on X_{0}(N) with p ∤ N. Thus assume, unless otherwise stated, that p ∤ N. Remember that T_{p} was defined to be the correspondence

\xymatrix{ & X_{0}(pN) \ar[dl]_{α} \ar[dr]^{β} \\ X_{0}(N) &  & X_{0}(N)}

Think of X_{0}(pN) as consisting of pairs (\underline{E}, D) where D is a cyclic subgroup of E of order p and \underline{E} is the enhanced elliptic curve consisting of an elliptic curve E along with a cyclic subgroup of order N. The degeneracy map α forgets the subgroup D and the degeneracy map β divides by it. By contravariant functoriality we have a commutative diagram

[figure: H^{0}(X_{0}(N), Ω^{1}); T_{p}^{∗}=α_{∗}∘β^{∗}; H^{0}(X_{0}(N), Ω^{1}); S_{2}(Γ_{0}(N)); T_{p}; S_{2}(Γ_{0}(N))]

Our convention to define T_{p}^{∗} as α_{∗} ∘ β^{∗} instead of β_{∗} ∘ α^{∗} was completely psychological because there is a canonical duality relating the two. We chose the way we did because of the analogy with the case of a morphism φ : Y → X with graph Γ which induces a correspondence

\xymatrix{ & Γ \ar[dl]_{π_{1}} \ar[dr]^{π_{2}} \\ Y &  & X}

Since the morphism φ induces a map on global sections in the other direction

H^{0}(X, Ω^{1}) = Γ(X) ⟶^{φ^{∗}} Γ(Y) = H^{0}(Y, Ω^{1})

it is psychologically natural for more general correspondence such as T_{p} to map from the right to the left.

The morphisms α and β in the definition of T_{p} are defined over Q. This can be seen using the general theory of representable functors. Thus since T_{p} is defined over Q most of the algebraic geometric objects we will construct related to T_{p} will be defined over Q.

12.5 Hecke operators acting on Jacobians

The Jacobian J(X_{0}(N)) = J_{0}(N) is an abelian variety defined over Q. There are both covariant and contravariant ways to construct J_{0}(N). Thus a map α :

X_{0}(pN) → X_{0}(N) induces maps

[figure: J_{0}(pN); J_{0}(pN); α_{∗}; J_{0}(N); α^{∗}; p+1; J_{0}(N)]

Note that α_{∗} ∘ α^{∗} : J_{0}(N) → J_{0}(N) is just multiplication by deg(α) = p + 1, since there are p + 1 subgroups of order p in \underline{E}. (At least when p ∤ N, when p|N there are only p subgroups.)

There are two possible ways to define T_{p} as an endomorphism of J_{0}(N). We could either define T_{p} as β_{∗} ∘ α^{∗} or equivalently as α_{∗} ∘ β^{∗} (assuming still that p ∤ N).

12.5.1 The Albanese Map

There is a way to map the curve X_{0}(N) into its Jacobian since the underlying group structure of J_{0}(N) is

J_{0}(N) = { divisors of degree 0 on X_{0}(N) } ; { principal divisors }

Once we have chosen a rational point, say ∞, on X_{0}(N) we obtain the Albanese map

θ : X_{0}(N) → J_{0}(N) : x ↦ x − ∞

which sends a point x to the divisor x − ∞. The map θ gives us a way to pullback differentials on J_{0}(N). Let Cot J_{0}(N) denote the cotangent space of J_{0}(N) (or the space of regular differentials). The diagram

\xymatrix{Cot J_{0}(N) \ar[d]^{≀}_{θ^{∗}} & Cot J_{0}(N) \ar[l]_{ξ_{p}^{∗}} \ar[d]^{θ^{∗}}_{≀} \\ H^{0}(X_{0}(N), Ω^{1}) & H^{0}(X_{0}(N), Ω^{1}) \ar[l]_{T_{p}^{∗}}}

may be taken to give a definition of ξ_{p} since there is a unique endomorphism ξ_{p} : J_{0}(N) → J_{0}(N) inducing a map ξ_{p}^{∗} which makes the diagram commute.

Now suppose Γ is a correspondence X ⇝ Y so we have a diagram

\xymatrix{ & Γ \ar[dl]_{α} \ar[dr]^{β} \\ X &  & Y}

For example, think of Γ as the graph of a morphism φ : X → Y. Then Γ should induce a natural map

H^{0}(Y, Ω^{1}) ⟶ H^{0}(X, Ω^{1}).

Taking Jacobians we see that the composition

J(X) ⟶^{α^{∗}} J(Γ) ⟶^{β_{∗}} J(Y)

gives a map β_{∗} ∘ α^{∗} : J(X) → J(Y). On cotangent spaces this induces a map

α^{∗} ∘ β_{∗} : H^{0}(Y, Ω^{1}) → H^{0}(X, Ω^{1}).

Now, after choice of a rational point, the map X → J(X) induces a map Cot J(X) → H^{0}(X, Ω^{1}). This is in fact independent of the choice of rational point since differentials on J(X) are invariant under translation.

The map J(X) → J(Y) is preferred in the literature. It is said to be induced by the Albanese functoriality of the Jacobian. We could have just as easily defined a map from J(Y) → J(X). To see this let

ψ = β_{∗} ∘ α^{∗} : J(X) → J(Y).

Dualizing induces a map ψ^{∨} = α_{∗} ∘ β^{∗}:

\xymatrix{J(X)^{∨} \ar[d]^{≅} & J(Y)^{∨} \ar[l]_{ψ^{∨}} \\ J(X) & J(Y) \ar[u]^{≅}}

Here we have used autoduality of Jacobians. This canonical duality is discussed in

[MFK94] and [Mum70] and in Milne’s article in [Sch65].

12.5.2 The Hecke algebra

We now have ξ_{p} = T_{p} ∈ End J_{0}(N) for every prime p. If p|N, then we must decide between α_{∗} ∘β^{∗} and β_{∗} ∘α^{∗}. The usual choice is the one which induces the usual T_{p} on cusp forms. If you don’t like your choice you can get out of it with Atkin-Lehner operators.

Let

T = Z[…, T_{p}, …] ⊂ End J_{0}(N)

then T is the same as T_{Z} ⊂ End(S_{2}(Γ_{0}(N))). To see this first note that there is a map T → T_{Z} which is not a prior injective, but which is injective because elements of End J_{0}(N) are completely determined by their action on Cot J_{0}(N).

12.6 The Eichler-Shimura relation

Suppose p ∤ N is a prime. The Hecke operator T_{p} and the Frobenius automorphism Frob_{p} induce, by functoriality, elements of End(J_{0}(N)_{F_{p}}), which we also denote T_{p} and Frob_{p}. The Eichler-Shimura relation asserts that the relation

T_{p} = Frob_{p} + Frob_{p}^{∨}  (12.6.1)

holds in End(J_{0}(N)_{F_{p}}). In this section we sketch the main idea behind why (12.6.1) holds. For more details and a proof of the analogous statement for J_{1}(N), see [Con01].^{2} 2

FIGURE 12.6.1. The reduction mod p of the Deligne-Rapoport model of X_{0}(Np)

Since J_{0}(N) is an abelian variety defined over Q, it is natural to ask for the primes p such that J_{0}(N) have good reduction. In the 1950s Igusa showed^{3} that 3 J_{0}(N) has good reduction for all p ∤ N. He viewed J_{0}(N) as a scheme over Spec(Q), then “spread things out” to make an abelian scheme over Spec(Z[1/N]). He did this by taking the Jacobian of the normalization of X_{0}(N) (which is defined over Z[1/N]) in P_{Z[1/N]}^{n}.

The Eichler-Shimura relation is a formula for T_{p} in characteristic p, or more precisely, for the corresponding endomorphisms in End(J_{0}(N)_{F_{p}})) for all p for which J_{0}(N) has good reduction at p. If p ∤ N, then X_{0}(N)_{F_{p}} has many of the same properties as X_{0}(N)_{Q}. In particular, the noncuspidal points on X_{0}(N)_{F_{p}} classify isomorphism classes of enhanced elliptic curves \underline{E} = (E, C), where E is an elliptic curve over F_{p} and C is a cyclic subgroup of E of order N. (Note that two pairs are considered isomorphic if they are isomorphic over \overline{F}_{p}.)

Next we ask what happens to the map T_{p} : J_{0}(N) → J_{0}(N) under reduction modulo p. To this end, consider the correspondence

\xymatrix{ & X_{0}(Np) \ar[dl]_{α} \ar[dr]^{β} \\ X_{0}(N) &  & X_{0}(N)}

that defines T_{p}. The curve X_{0}(N) has good reduction at p, but X_{0}(Np) typically does not. Deligne and Rapaport [DR73] showed that X_{0}(Np) has relatively benign reduction at p. Over F_{p}, the reduction X_{0}(Np)_{F_{p}} can be viewed as two copies of X_{0}(N) glued at the supersingular points, as illustrated in Figure 12.6.1.

The set of supersingular points

Σ ⊂ X_{0}(N)(\overline{F}_{p})

is the set of points in X_{0}(N) represented by pairs \underline{E} = (E, C), where E is a supersingular elliptic curve (so E(\overline{F}_{p})[p] = 0). There are exactly g+1 supersingular points, where g is the genus of X_{0}(N).^{4} 4

Consider the correspondence T_{p} : X_{0}(N) ⇝ X_{0}(N) which takes an enhanced elliptic curve \underline{E} to the sum ∑ \underline{E}/D of all quotients of \underline{E} by subgroups D of order p. This is the correspondence

\xymatrix{ & X_{0}(pN) \ar[dl]_{α} \ar[dr]^{β} \\ X_{0}(N) &  & X_{0}(N),}  (12.6.2)

where the map α forgets the subgroup of order p, and β quotients out by it. From this one gets T_{p} : J_{0}(N) → J_{0}(N) by functoriality. Remark 12.6.1. There are many ways to think of J_{0}(N). The cotangent space Cot J_{0}(N) of J_{0}(N) is the space of holomorphic (or translation invariant) differentials on J_{0}(N), which is isomorphic to S_{2}(Γ_{0}(N)). This gives a connection between our geometric definition of T_{p} and the definition, presented earlier,^{5} of T_{p} as an 5 operator on a space of cusp forms.

2Add more references to original source materials...

3Ken, what’s a reference for this?

4Reference.

The Eichler-Shimura relation takes place in End(J_{0}(N)_{F_{p}}). Since X_{0}(N) reduces “nicely” in characteristic p, we can apply the Jacobian construction to X_{0}(N)_{F_{p}}.

Lemma 12.6.2. The natural reduction map

End(J_{0}(N)) ↪ End(J_{0}(N)_{F_{p}})

is injective.

Proof. Let ℓ ∤ Np be a prime. By [ST68, Thm. 1, Lem. 2], the reduction to characteristic p map induces an isomorphism

J_{0}(N)(\overline{Q})[ℓ^{∞}] ≅ J_{0}(N)(\overline{F}_{p})[ℓ^{∞}].

If φ ∈ End(J_{0}(N)) reduces to the 0 map in End(J_{0}(N)_{F_{p}}), then J_{0}(N)(\overline{Q})[ℓ^{∞}] must be contained in ker(φ). Thus φ induces the 0 map on Tate_{ℓ}(J_{0}(N)), so φ = 0.

Let F : X_{0}(N)_{F_{p}} → X_{0}(N)_{F_{p}} be the Frobenius map in characteristic p. Thus, if K = K(X_{0}(N)) is the function field of the nonsingular curve X_{0}(N), then F : K → K is induced by the pth power map a ↦ a^{p}. Remark 12.6.3. The Frobenius map corresponds to the pth powering map on points. For example, if X = Spec(F_{p}[t]), and z = (Spec(\overline{F}_{p}) → X) is a point defined by a homomorphism α : F_{p}[t] ↦ \overline{F}_{p}, then F (z) is the composite

F_{p}[t] ⟶^{x↦x^{p}} F_{p}[t] ⟶^{α} \overline{F}_{p}.

If α(t) = ξ, then F (z)(t) = α(t^{p}) = ξ^{p}.

By both functorialities, F induces maps on the Jacobian of X_{0}(N)_{F_{p}}:

Frob_{p} = F_{∗} and Ver_{p} = Frob_{p}^{∨} = F^{∗},

which we illustrate as follows:

\xymatrix{J_{0}(N)_{F_{p}} \ar[r]^{^{Ver^{p}}} & J_{0}(N)_{F_{p}} \ar[l]^{_{Frob_{p}}}}

Note that Ver_{p} ∘ Frob_{p} = Frob_{p} ∘ Ver_{p} = [p] since p is the degree of F (for example, if K = F_{p}(t), then F (K) = F_{p}(t^{p}) is a subfield of degree p, so the map induced by F has degree p).

Theorem 12.6.4 (Eichler-Shimura Relation). Let N be a positive integer and p ∤ N be a prime. Then the following relation holds:

T_{p} = Frob_{p} + Ver_{p} ∈ End(J_{0}(N)_{F_{p}}).

5more precise

Sketch of Proof. We view X_{0}(pN)_{F_{p}} as two copies of X_{0}(N)_{F_{p}} glued along corresponding supersingular points Σ, as in Figure 12.6.1. This diagram and the correspondence (12.6.2) that defines T_{p} translate into the following diagram of schemes over F_{p}:

\xymatrix{ & Σ \ar@{^{(}->}[dl] \ar@{^{(}->}[dr] \\ X_{0}(N)_{F_{p}} \ar@{^{(}->}[dr]^{r} \ar[dd]^{≅} &  & X_{0}(N)_{F_{p}} \ar@{^{(}->}[dl]_{s} \ar[dd]^{≅} \\  & X_{0}(pN)_{F_{p}} \ar[dl]_{α} \ar[dr]^{β} \\ X_{0}(N)_{F_{p}} &  & X_{0}(N)_{F_{p}}}

The maps r and s are defined as follows. Recall that a point of X_{0}(N)_{F_{p}} is an enhanced elliptic curve \underline{E} = (E, C) consisting of an elliptic curve E (not necessarily defined over F_{p}) along with a cyclic subgroup C of order N. We view a point on X_{0}(Np) as a triple (E, C, E → E′), where (E, C) is as above and E → E′ is an isogeny of degree p. We use an isogeny instead of a cyclic subgroup of order p because E(\overline{F}_{p})[p] has order either 1 or p, so the data of a cyclic subgroup of order p holds very little information.

The map r sends \underline{E} to (\underline{E}, φ), where φ is the isogeny of degree p,

φ : E ⟶^{Frob} E^{(p)}.

Here E^{(p)} is the curve obtained from E by hitting all defining equations by Frobenious, that is, by pth powering the coefficients of the defining equations for E. We introduce E^{(p)} since if E is not defined over F_{p}, then Frobenious does not define an endomorphism of E. Thus r is the map

r : \underline{E} ↦ (\underline{E}, E ⟶^{Frob_{p}} E^{(p)}),

and similarly we define s to be the map

s : \underline{E} ↦ (E^{(p)}, C, E ⟵^{Ver_{p}} E^{(p)})

where Ver_{p} is the dual of Frob_{p} (so Ver_{p} ∘ Frob_{p} = Frob_{p} ∘ Ver_{p} = [p]).

We view α as the map sending (\underline{E}, E → E′) to \underline{E}, and similarly we view β as the map sending (\underline{E}, E → E′) to the pair (E′, C′), where C′ is the image of C in E′ via E → E′. Thus

α : (E → E′) ↦ E
β : (E′ → E) ↦ E′

It now follows immediately that α ∘ r = id and β ∘ s = id. Note also that α ∘ s = β ∘ r = F is the map E ↦ E^{(p)}.

Away from the finitely many supersingular points, we may view X_{0}(pN)_{F_{p}} as the disjoint union of two copies of X_{0}(N)_{F_{p}}. Thus away from the supersingular points, we have the following equality of correspondences:

[figure: X_{0}(pN)_{F_{p}}; α; β; X_{0}(N)_{F_{p}}; X_{0}(N)_{F_{p}}; =′; X_{0}(N)_{F_{p}}; id=α∘r; F =β∘r; X_{0}(N)_{F_{p}}; X_{0}(N)_{F_{p}}; +; X_{0}(N)_{F_{p}}; F =α∘s; id=β∘s; X_{0}(N)_{F_{p}},; X_{0}(N)_{F_{p}}]

where F = Frob_{p}, and the =′ means equality away from the supersingular points. Note that we are simply “pulling back” the correspondence; in the first summand we use the inclusion r, and in the second we use the inclusion s.

This equality of correspondences implies that the equality

T_{p} = Frob_{p} + Ver_{p}

of endomorphisms holds on a dense subset of J_{0}(N)_{F_{p}}, hence on all J_{0}(N)_{F_{p}}.

12.7 Applications of the Eichler-Shimura relation

12.7.1 The Characteristic polynomial of Frobenius

How can we apply the relation T_{p} = Frob + Ver in End(J_{0}(N)_{F_{p}})? Let ℓ ∤ pN be a prime and consider the ℓ-adic Tate module

Tate_{ℓ}(J_{0}(N)) = (\underleftarrow{lim} J_{0}(N)[ℓ^{ν}]) ⊗_{Z_{ℓ}} Q_{ℓ}

which is a vector space of dimension 2g over Q_{ℓ}, where g is the genus of X_{0}(N) or the dimension of J_{0}(N). Reduction modulo p induces an isomorphism

Tate_{ℓ}(J_{0}(N)) → Tate_{ℓ}(J_{0}(N)_{F_{p}})

(see the proof of Lemma 12.6.2). On Tate_{ℓ}(J_{0}(N)_{F_{p}}) we have linear operators Frob_{p}, Ver_{p} and T_{p} which, as we saw in Section 12.6, satisfy

Frob_{p} + Ver_{p} = T_{p},  and
Frob_{p} ∘ Ver_{p} = Ver_{p} ∘ Frob_{p} = [p].

The endomorphism [p] is invertible on Tate_{ℓ}(J_{0}(N)_{F_{p}}), since p is prime to ℓ, so Ver_{p} and Frob_{p} are also invertible and

T_{p} = Frob_{p} +[p] Frob_{p}^{−1} .

Multiplying both sides by Frob_{p} and rearranging, we see that

Frob_{p}^{2} −T_{p} Frob_{p} +[p] = 0 ∈ End(Tate_{ℓ}(J_{0}(N)_{F_{p}})).

This is a beautiful quadratic relation, so we should be able to get something out of it. We will come back to this shortly, but first we consider the various objects acting on the ℓ-adic Tate module.

The module Tate_{ℓ}(J_{0}(N)) is acted upon in a natural way by 1. The Galois group Gal(\overline{Q}/Q) of Q, and

2. End_{Q}(J_{0}(N)) ⊗_{Z_{ℓ}} Q_{ℓ} (which acts by functoriality).

These actions commute with each other since endomorphisms defined over Q are not affected by the action of Gal(\overline{Q}/Q). Reducing modulo p, we also have the following commuting actions:

3. The Galois group Gal(\overline{F}_{p}/F_{p}) of F_{p}, and

4. End_{F_{p}}(J_{0}(N)) ⊗_{Z_{ℓ}} Q_{ℓ}.

Note that a decomposition group group D_{p} ⊂ Gal(\overline{Q}/Q) acts, after quotienting out by the corresponding inertia group, in the same way as Gal(\overline{F}_{p}/F_{p}) and the action is unramified, so action 3 is a special case of action 1.

The Frobenius elements φ_{p} ∈ Gal(\overline{F}_{p}/F_{p}) and Frob_{∈} End_{F_{p}}(J_{0}(N)) ⊗_{Z_{ℓ}} Q_{ℓ} induce the same operator on Tate_{ℓ}(J_{0}(N)_{F_{p}}). Note that while φ_{p} naturally lives in a quotient of a decomposition group, one often takes a lift to get an element in Gal(\overline{Q}/Q).

On Tate_{ℓ}(J_{0}(N)_{F_{p}}) we have a quadratic relationship

φ_{p}^{2} − T_{p}φ_{p} + p = 0.

This relation plays a role when one separates out pieces of J_{0}(N) in order to construct Galois representations attached to newforms of weight 2. Let

R = Z[…, T_{p}, …] ⊂ End J_{0}(N),

where we only adjoin those T_{p} with p ∤ N. Think of R as a reduced Hecke algebra; in particular, R is a subring of T. Then

R ⊗ Q = ∏_{i=1}^{r} E_{i},

where the E_{i} are totally real number fields. The factors E_{i} are in bijection with the Galois conjugacy classes of weight 2 newforms f on Γ_{0}(M) (for some M|N). The bijection is the map

f ↦ Q(coefficients of f) = E_{i}

Observe that the map is the same if we replace f by one of its conjugates. This decomposition is a decomposition of a subring

R ⊗ Q ⊂ End(J_{0}(N)) ⊗ Q =^{def} End(J_{0}(N) ⊗ Q).

Thus it induces a direct product decomposition of J_{0}(N), so J_{0}(N) gets divided up into subvarieties which correspond to conjugacy classes of newforms.

The relationship

φ_{p}^{2} − T_{p}φ_{p} + p = 0  (12.7.1)

suggests that

tr(φ_{p}) = T_{p}  and det φ_{p} = p.  (12.7.2)

This is true, but (12.7.2) does not follow formally just from the given quadratic relation. It can be proved by combining (12.7.1) with the Weil pairing.

12.7 Applications of the Eichler-Shimura relation 107

12.7.2 The Cardinality of J_{0}(N)(F_{p})

Proposition 12.7.1. Let p ∤ N be a prime, and let f be the characteristic polynomial of T_{p} acting on S_{2}(Γ_{0}(N)). Then

#J_{0}(N)(F_{p}) = f(p + 1).

6

6

6Add details later, along with various generalizations.

108 12. Hecke Operators as Correspondences

Abelian Varieties

This chapter provides foundational background about abelian varieties and Jacobians, with an aim toward what we will need later when we construct abelian varieties attached to modular forms. We will not give complete proofs of very much, but will try to give precise references whenever possible, and many examples.

We will follow the articles by Rosen [Ros86] and Milne [Mil86] on abelian varieties. We will try primarily to explain the statements of the main results about abelian varieties, and prove results when the proofs are not too technical and enhance understanding of the statements.

13.1 Abelian varieties

Definition 13.1.1 (Variety). A variety X over a field k is a finite-type separated scheme over k that is geometrically integral.

The condition that X be geometrically integral means that X\overline{k} is reduced (no nilpotents in the structure sheaf) and irreducible.

Definition 13.1.2 (Group variety). A group variety is a group object in the category of varieties. More precisely, a group variety X over a field k is a variety equipped with morphisms

m : X × X → X and i : X → X

and a point 1_{X} ∈ A(k) such that m, i, and 1_{X} satisfy the axioms of a group; in particular, for every k-algebra R they give X(R) a group structure that depends in a functorial way on R.

Definition 13.1.3 (Abelian Variety). An abelian variety A over a field k is a complete group variety.

Theorem 13.1.4. Suppose A is an abelian variety. Then 1. The group law on A is commutative.

2. A is projective, i.e., there is an embedding from A into P^{n} for some n.

3. If k = C, then A(k) is analytically isomorphic to V/L, where V is a finitedimensional complex vector space and L is a lattice in V. (A lattice is a free Z-module of rank equal to 2 dim V such that RL = V.)

Proof. Part 1 is not too difficult, and can be proved by showing that every morphism of abelian varieties is the composition of a homomorphism with a translation, then applying this result to the inversion map (see [Mil86, Cor. 2.4]). Part 2 is proved with some effort in [Mil86, §7]. Part 3 is proved in [Mum70, §I.1] using the exponential map from Lie theory from the tangent space at 0 to A.

13.2 Complex tori

Let A be an abelian variety over C. By Theorem 13.1.4, there is a complex vector space V and a lattice L in V such that A(C) = V/L, that is to say, A(C) is a complex torus.

More generally, if V is any complex vector space and L is a lattice in V, we call the quotient T = V/L a complex torus. In this section, we prove some results about complex tori that will help us to understand the structure of abelian varieties, and will also be useful in designing algorithms for computing with abelian varieties.

The differential 1-forms and first homology of a complex torus are easy to understand in terms of T. If T = V/L is a complex torus, the tangent space to 0 ∈ T is canonically isomorphic to V. The C-linear dual V^{∗} = Hom_{C}(V, C) is isomorphic to the C-vector space Ω(T) of holomorphic differential 1-forms on T. Since V → T is the universal covering of T, the first homology H_{1}(T, Z) of T is canonically isomorphic to L.

13.2.1 Homomorphisms

Suppose T_{1} = V_{1}/L_{1} and T_{2} = V_{2}/L_{2} are two complex tori. If φ : T_{1} → T_{2} is a (holomorphic) homomorphism, then φ induces a C-linear map from the tangent space of T_{1} at 0 to the tangent space of T_{2} at 0. The tangent space of T_{i} at 0 is canonically isomorphic to V_{i}, so φ induces a C-linear map V_{1} → V_{2}. This maps sends L_{1} into L_{2}, since L_{i} = H_{1}(T_{i}, Z). We thus have the following diagram:

\xymatrix{0 \ar[d] & 0 \ar[d] \\ L_{1} \ar[r]^{ρ_{Z}(φ)} \ar[d] & L_{2} \ar[d] \\ V_{1} \ar[r]^{ρ_{C}(φ)} \ar[d] & L_{2} \ar[d] \\ T_{1} \ar[r]^{φ} \ar[d] & T_{2} \ar[d] \\ 0 & 0}  (13.2.1)

We obtain two faithful representations of Hom(T_{1}, T_{2}),

ρ_{C} : Hom(T_{1}, T_{2}) → Hom_{C}(V_{1}, V_{2})
ρ_{Z} : Hom(T_{1}, T_{2}) → Hom_{Z}(L_{1}, L_{2}).

Suppose ψ ∈ Hom_{Z}(L_{1}, L_{2}). Then ψ = ρ_{Z}(φ) for some φ ∈ Hom(T_{1}, T_{2}) if and only if there is a complex linear homomorphism f : V_{1} → V_{2} whose restriction to L_{1} is ψ. Note that f = ψ ⊗ R is uniquely determined by ψ, so ψ arises from some φ precisely when f is C-linear. This is the case if and only if fJ_{1} = J_{2}f, where J_{n} : V_{n} → V_{n} is the R-linear map induced by multiplication by i = \sqrt{−1} ∈ C.

Example 13.2.1.

1. Suppose L_{1} = Z + Zi ⊂ V_{1} = C. Then with respect to the basis 1, i, we have J_{1} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). One finds that Hom(T_{1}, T_{1}) is the free Z-module of rank 2 whose image via ρ_{Z} is generated by J_{1} and ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). As a ring Hom(T_{1}, T_{1}) is isomorphic to Z[i].

2. Suppose L_{1} = Z + Zαi ⊂ V_{1} = C, with α^{3} = 2. Then with respect to the basis 1, αi, we have J_{1} = ( \begin{matrix}0 & −α \\ 1/α & 0 \end{matrix} ). Only the scalar integer matrices commute with J_{1}.

Proposition 13.2.2. Let T_{1} and T_{2} be complex tori. Then Hom(T_{1}, T_{2}) is a free Z-module of rank at most 4 dim T_{1} · dim T_{2}.

Proof. The representation ρ_{Z} is faithful (injective) because φ is determined by its action on L_{1}, since L_{1} spans V_{1}. Thus Hom(T_{1}, T_{2}) is isomorphic to a subgroup of Hom_{Z}(L_{1}, L_{2}) ≅ Z^{d}, where d = 2 dim V_{1} · 2 dim V_{2}.

Lemma 13.2.3. Suppose φ : T_{1} → T_{2} is a homomorphism of complex tori. Then the image of φ is a subtorus of T_{2} and the connected component of ker(φ) is a subtorus of T_{1} that has finite index in ker(φ).

Proof. Let W = ker(ρ_{C}(φ)). Then the following diagram, which is induced by φ, has exact rows and columns:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & L_{1} ∩ W \ar[d] \ar[r] & L_{1} \ar[r] \ar[d] & L_{2} \ar[r] \ar[d] & L_{2}/φ(L_{1}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W \ar[d] \ar[r] & V_{1} \ar[r] \ar[d] & V_{2} \ar[r] \ar[d] & V_{2}/φ(V_{1}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ker(φ) \ar[r] & T_{1} \ar[r] \ar[d] & T_{2} \ar[r] \ar[d] & T_{2}/φ(T_{1}) \ar[r] \ar[d] & 0 \\  &  & 0 & 0 & 0}

Using the snake lemma, we obtain an exact sequence

0 → L_{1} ∩ W → W → ker(φ) → L_{2}/φ(L_{1}) → V_{2}/φ(V_{1}) → T_{2}/φ(T_{1}) → 0.

Note that T_{2}/φ(T_{1}) is compact because it is the continuous image of a compact set, so the cokernel of φ is a torus (it is given as a quotient of a complex vector space by a lattice).

The kernel ker(φ) ⊂ T_{1} is a closed subset of the compact set T_{1}, so is compact. Thus L_{1}∩W is a lattice in W. The map L_{2}/φ(L_{1}) → V_{2}/φ(V_{1}) has kernel generated by the saturation of φ(L_{1}) in L_{2}, so it is finite, so the torus W/(L_{1} ∩ W) has finite index in ker(φ).

Remark 13.2.4. The category of complex tori is not an abelian category because kernels need not be in the category.

13.2.2 Isogenies

Definition 13.2.5 (Isogeny). An isogeny φ : T_{1} → T_{2} of complex tori is a surjective morphism with finite kernel. The degree deg(φ) of φ is the order of the kernel of φ.

Note that deg(φ ∘ φ′) = deg(φ) deg(φ′).

Lemma 13.2.6. Suppose that φ is an isogeny. Then the kernel of φ is isomorphic to the cokernel of ρ_{Z}(φ).

Proof. (This is essentially a special case of Lemma 13.2.3.) Apply the snake lemma to the morphism (13.2.1) of short exact sequences, to obtain a six-term exact sequence

0 → K_{L} → K_{V} → K_{T} → C_{L} → C_{V} → C_{T} → 0,

where K_{X} and C_{X} are the kernel and cokernel of X_{1} → X_{2}, for X = L, V, T, respectively. Since φ is an isogeny, the induced map V_{1} → V_{2} must be an isomorphism, since otherwise the kernel would contain a nonzero subspace (modulo a lattice), which would be infinite. Thus K_{V} = C_{V} = 0. It follows that K_{T} ≅ C_{L}, as claimed.

One consequence of the lemma is that if φ is an isogeny, then

deg(φ) = [L_{1} : ρ_{Z}(φ)(L_{1})] = | det(ρ_{Z}(φ))|.

Proposition 13.2.7. Let T be a complex torus of dimension d, and let n be a positive integer. Then multiplication by n, denoted [n], is an isogeny T → T with kernel T [n] ≅ (Z/nZ)^{2d} and degree n^{2d}.

Proof. By Lemma 13.2.6, T [n] is isomorphic to L/nL, where T = V/L. Since L ≈ Z^{2d}, the proposition follows.

We can now prove that isogeny is an equivalence relation.

Proposition 13.2.8. Suppose φ : T_{1} → T_{2} is a degree m isogeny of complex tori of dimension d. Then there is a unique isogeny φ̂ : T_{2} → T_{1} of degree m^{2d−1} such that φ̂ ∘ φ = φ ∘ φ̂ = [m].

Proof. Since ker(φ) ⊂ ker([m]), the map [m] factors through φ, so there is a morphism φ̂ such that φ̂ ∘ φ = [m]:

\xymatrix{T_{1} \ar[r]^{φ} \ar[dr]_{[m]} & T_{2} \ar[d]^{φ̂} \\  & T_{1}}

We have

(φ ∘ φ̂ − [m]) ∘ φ = φ ∘ φ̂ ∘ ϕ − [m] ∘ φ = φ ∘ φ̂ ∘ ϕ − φ ∘ [m] = φ ∘ (φ̂ ∘ ϕ − [m]) = 0.

This implies that φ ∘ φ̂ = [m], since φ is surjective. Uniqueness is clear since the difference of two such morphisms would vanish on the image of φ. To see that φ̂ has degree m^{2d−1}, we take degrees on both sides of the equation φ̂ ∘ φ = [m].

13.2.3 Endomorphisms

The ring End(T) = Hom(T, T) is called the endomorphism ring of the complex torus T. The endomorphism algebra of T is End_{0}(T) = End(T) ⊗_{Z} Q.

Definition 13.2.9 (Characteristic polynomial). The characteristic polynomial of φ ∈ End(T) is the characteristic polynomial of the ρ_{Z}(φ). Thus the characteristic polynomial is a monic polynomial of degree 2 dim T.

13.3 Abelian varieties as complex tori

In this section we introduce extra structure on a complex torus T = V/L that will enable us to understand whether or not T is isomorphic to A(C), for some abelian variety A over C. When dim T = 1, the theory of the Weierstrass ℘ function implies that T is always E(C) for some elliptic curve. In contrast, the generic torus of dimension > 1 does not arise from an abelian variety.

In this section we introduce the basic structures on complex tori that are needed to understand which tori arise from abelian varieties, to construct the dual of an abelian variety, to see that End_{0}(A) is a semisimple Q-algebra, and to understand the polarizations on an abelian variety. For proofs, including extensive motivation from the one-dimensional case, read the beautifully written book [SD74] by Swinnerton-Dyer, and for another survey that strongly influenced the discussion below, see Rosen’s [Ros86].

13.3.1 Hermitian and Riemann forms

Let V be a finite-dimensional complex vector space.

Definition 13.3.1 (Hermitian form). A Hermitian form is a conjugate-symmetric pairing

H : V × V → C

that is C-linear in the first variable and C-antilinear in the second. Thus H is R-bilinear, H(iu, v) = iH(u, v) = H(u, \overline{i}v), and H(u, v) = \overline{H(v, u)}.

Write H = S + iE, where S, E : V × V → R are real bilinear pairings.

Proposition 13.3.2. Let H, S, and E be as above.

1. We have that S is symmetric, E is antisymmetric, and

S(u, v) = E(iu, v), S(iu, iv) = S(u, v), E(iu, iv) = E(u, v).

2. Conversely, if E is a real-valued antisymmetric bilinear pairing on V such that E(iu, iv) = E(u, v), then H(u, v) = E(iu, v) + iE(u, v) is a Hermitian form on V. Thus there is a bijection between the Hermitian forms on V and the real, antisymmetric bilinear forms E on V such that E(iu, iv) = E(u, v).

Proof. To see that S is symmetric, note that 2S = H +\overline{H} and H +\overline{H} is symmetric because H is conjugate symmetric. Likewise, E = (H − \overline{H})/(2i), so

E(v, u) = \frac{1}{2i} (H(v, u) − \overline{H(v, u)}) = \frac{1}{2i} (\overline{H(u, v)} − H(u, v)) = −E(u, v),

which implies that E is antisymmetric. To see that S(u, v) = E(iu, v), rewrite both S(u, v) and E(iu, v) in terms of H and simplify to get an identity. The other two identities follow since

H(iu, iv) = iH(u, iv) = i\overline{i}H(u, v) = H(u, v).

Suppose E : V × V → R is as in the second part of the proposition. Then

H(iu, v) = E(i^{2}u, v) + iE(iu, v) = −E(u, v) + iE(iu, v) = iH(u, v),

and the other verifications of linearity and antilinearity are similar. For conjugate symmetry, note that

H(v, u) = E(iv, u) + iE(v, u) = −E(u, iv) − iE(u, v)
= −E(iu, −v) − iE(u, v) = H(u, v).

Note that the set of Hermitian forms is a group under addition.

Definition 13.3.3 (Riemann form). A Riemann form on a complex torus T = V/L is a Hermitian form H on V such that the restriction of E = Im(H) to L is integer valued. If H(u, u) ≥ 0 for all u ∈ V then H is positive semi-definite and if H is positive and H(u, u) = 0 if and only if u = 0, then H is nondegenerate.

Theorem 13.3.4. Let T be a complex torus. Then T is isomorphic to A(C), for some abelian variety A, if and only if there is a nondegenerate Riemann form on T.

This is a nontrivial theorem, which we will not prove here. It is proved in [SD74, Ch.2] by defining an injective map from positive divisors on T = V/L to positive semi-definite Riemann forms, then constructing positive divisors associated to theta functions on V. If H is a nondegenerate Riemann form on T, one computes the dimension of a space of theta functions that corresponds to H in terms of the determinant of E = Im(H). Since H is nondegenerate, this space of theta functions is nonzero, so there is a corresponding nondegenerate positive divisor D. Then a basis for

L(3D) = {f : (f) + 3D is positive } ∪ {0}

determines an embedding of T in a projective space.

Why the divisor 3D instead of D above? For an elliptic curve y^{2} = x^{3} + ax + b, we could take D to be the point at infinity. Then L(3D) consists of the functions with a pole of order at most 3 at infinity, which contains 1, x, and y, which have poles of order 0, 2, and 3, respectively. Remark 13.3.5. (Copied from page 39 of [SD74].) When n = dim V > 1, however, a general lattice L will admit no nonzero Riemann forms. For if λ_{1}, …, λ_{2n} is a base for L then E as an R-bilinear alternating form is uniquely determined by the E(λ_{i}, λ_{j}), which are integers; and the condition E(z, w) = E(iz, iw) induces linear relations with real coefficients between E(λ_{i}, λ_{j}), which for general L have no nontrivial integer solutions.

13.3.2 Complements, quotients, and semisimplicity of the endomorphism algebra

Lemma 13.3.6. If T possesses a nondegenerate Riemann form and T′ ⊂ T is a subtorus, then T′ also possesses a nondegenerate Riemann form.

Proof. If H is a nondegenerate Riemann form on a torus T and T′ is a subtorus of T, then the restriction of H to T′ is a nondegenerate Riemann form on T′ (the restriction is still nondegenerate because H is positive definite).

Lemma 13.3.6 and Lemma 13.2.3 together imply that the kernel of a homomorphism of abelian varieties is an extension of an abelian variety by a finite group.

Lemma 13.3.7. If T possesses a nondegenerate Riemann form and T → T′ is an isogeny, then T′ also possesses a nondegenerate Riemann form.

Proof. Suppose T = V/L and T′ = V′/L′. Since the isogeny is induced by an isomorphism V → V′ that sends L into L′, we may assume for simplicity that V = V′ and L ⊂ L′. If H is a nondegenerate Riemann form on V/L, then E = Re(H) need not be integer valued on L′. However, since L has finite index in L′, there is some integer d so that dE is integer valued on L′. Then dH is a nondegenerate Riemann form on V/L′.

Note that Lemma 13.3.7 implies that the quotient of an abelian variety by a finite subgroup is again an abelian variety.

Theorem 13.3.8 (Poincare Reducibility). Let A be an abelian variety and suppose A′ ⊂ A is an abelian subvariety. Then there is an abelian variety A′′ ⊂ A such that A = A′ + A′′ and A′ ∩ A′′ is finite. (Thus A is isogenous to A′ × A′′.)

Proof. We have A(C) ≈ V/L and there is a nondegenerate Riemann form H on V/L. The subvariety A′ is isomorphic to V′/L′, where V′ is a subspace of V and L′ = V′ ∩ L. Let V′′ be the orthogonal complement of V′ with respect to H, and let L′′ = L∩V′′. To see that L′′ is a lattice in V′′, it suffices to show that L′′ is the orthogonal complement of L′ in L with respect to E = Im(H), which, because E is integer valued, will imply that L′′ has the correct rank. First, suppose that v ∈ L′′; then, by definition, v is in the orthogonal complement of L′ with respect to H, so for any u ∈ L′, we have 0 = H(u, v) = S(u, v) + iE(u, v), so E(u, v) = 0. Next, suppose that v ∈ L satisfies E(u, v) = 0 for all u ∈ L′. Since V′ = RL′ and E is R-bilinear, this implies E(u, v) = 0 for any u ∈ V′. In particular, since V′ is a complex vector space, if u ∈ L′, then S(u, v) = E(iu, v) = 0, so H(u, v) = 0.

We have shown that L′′ is a lattice in V′′, so A′′ = V′′/L′′ is an abelian subvariety of A. Also L′+L′′ has finite index in L, so there is an isogeny V′/L′⊕V′′/L′′ → V/L induced by the natural inclusions.

Proposition 13.3.9. Suppose A′ ⊂ A is an inclusion of abelian varieties. Then the quotient A/A′ is an abelian variety.

Proof. Suppose A = V/L and A′ = V′/L′, where V′ is a subspace of V. Let W = V/V′ and M = L/(L ∩ V′). Then, W/M is isogenous to the complex torus V′′/L′′ of Theorem 13.3.8 via the natural map V′′ → W. Applying Lemma 13.3.7 completes the proof.

Definition 13.3.10. An abelian variety A is simple if it has no nonzero proper abelian subvarieties.

Proposition 13.3.11. The algebra End_{0}(A) is semisimple.

Proof. Using Theorem 13.3.8 and induction, we can find an isogeny

A ≃ A_{1}^{n_{1}} × A_{2}^{n_{2}} × ⋯ × A_{r}^{n_{r}}

with each A_{i} simple. Since End_{0}(A) = End(A) ⊗ Q is unchanged by isogeny, and Hom(A_{i}, A_{j}) = 0 when i ≠ j, we have

End_{0}(A) = End_{0}(A_{1}^{n_{1}}) × End_{0}(A_{2}^{n_{2}}) × ⋯ × End_{0}(A_{r}^{n_{r}})

Each of End_{0}(A_{i}^{n_{i}}) is isomorphic to M_{n_{i}}(D_{i}), where D_{i} = End_{0}(A_{i}). By Schur’s Lemma, D_{i} = End_{0}(A_{i}) is a division algebra over Q (proof: any nonzero endomorphism has trivial kernel, and any injective linear transformation of a Q-vector space is invertible), so End_{0}(A) is a product of matrix algebras over division algebras over Q, which proves the proposition.

13.3.3 Theta functions

Suppose T = V/L is a complex torus.

Definition 13.3.12 (Theta function). Let M : V × L → C and J : L → C be set-theoretic maps such that for each λ ∈ L the map z ↦ M(z, λ) is C-linear. A theta function of type (M, J) is a function θ : V → C such that for all z ∈ V and λ ∈ L, we have

θ(z + λ) = θ(z) · exp(2πi(M(z, λ) + J(λ))).

Suppose that θ(z) is a nonzero holomorphic theta function of type (M, J). The M(z, λ), for various λ, cannot be unconnected. Let F (z, λ) = 2πi(M(z, λ)+J(λ)).

Lemma 13.3.13. For any λ, λ′ ∈ L, we have

F (z, λ + λ′) = F (z + λ, λ′) + F (z, λ) (mod 2πi).

Thus

M(z, λ + λ′) = M(z, λ) + M(z, λ′),  (13.3.1)

and

J(λ + λ′) − J(λ) − J(λ′) ≡ M(λ, λ′) (mod Z).

Proof. Page 37 of [SD74].

Using (13.3.1) we see that M extends uniquely to a function M̃ : V × V → C which is C-linear in the first argument and R-linear in the second. Let

E(z, w) = M̃(z, w) − M(w, z),
H(z, w) = E(iz, w) + iE(z, w).

Proposition 13.3.14. The pairing H is Riemann form on T with real part E.

We call H the Riemann form associated to θ.

13.4 A Summary of duality and polarizations

Suppose A is an abelian variety over an arbitrary field k. In this section we summarize the most important properties of the dual abelian variety A^{∨} of A. First we review the language of sheaves on a scheme X, and define the Picard group of X as the group of invertible sheaves on X. The dual of A is then a variety whose points correspond to elements of the Picard group that are algebraically equivalent to 0. Next, when the ground field is C, we describe how to view A^{∨} as a complex torus in terms of a description of A as a complex torus. We then define the Néron-Severi group of A and relate it to polarizations of A, which are certain homomorphisms A → A^{∨}. Finally we observe that the dual is functorial.

13.4.1 Sheaves

We will use the language of sheaves, as in [Har77], which we now quickly recall. A pre-sheaf of abelian groups F on a scheme X is a contravariant functor from the category of open sets on X (morphisms are inclusions) to the category of abelian groups. Thus for every open set U ⊂ X there is an abelian group F(U), and if U ⊂ V, then there is a restriction map F(V) → F(U). (We also require that F(∅) = 0, and the map F(U) → F(U) is the identity map.) A sheaf is a pre-sheaf whose sections are determined locally (for details, see [Har77, §II.1]).

Every scheme X is equipped with its structure sheaf O_{X}, which has the property that if U = Spec(R) is an affine open subset of X, then O_{X}(U) = R. A sheaf of O_{X}-modules is a sheaf M of abelian groups on X such that each abelian group has the structure of O_{X}-module, such that the restriction maps are module morphisms. A locally-free sheaf of O_{X}-modules is a sheaf M of O_{X}-modules, such that X can be covered by open sets U so that M|_{U} is a free O_{X}-module, for each U.

13.4.2 The Picard group

An invertible sheaf is a sheaf L of O_{X}-modules that is locally free of rank 1. If L is an invertible sheaf, then the sheaf-theoretic Hom, L^{∨} = Hom(L, O_{X}) has the property that L^{∨}⊗L = O_{X}. The group Pic(X) of invertible sheaves on a scheme X is called the Picard group of X. See [Har77, §II.6] for more details.

Let A be an abelian variety over a field k. An invertible sheaf L on A is algebraically equivalent to 0 if there is a connected variety T over k, an invertible sheaf M on A ×_{k} T, and t_{0}, t_{1} ∈ T (k) such that M_{t_{0}} ≅ L and M_{t_{1}} ≅ O_{A}. Let Pic^{0}(A) be the subgroup of elements of Pic(A) that are algebraically equivalent to 0.

The dual A^{∨} of A is a (unique up to isomorphism) abelian variety such that for every field F that contains the base field k, we have A^{∨}(F) = Pic^{0}(A_{F}). For the precise definition of A^{∨} and a proof that A^{∨} exists, see [Mil86, §9–10].

13.4.3 The Dual as a complex torus

When A is defined over the complex numbers, so A(C) = V/L for some vector space V and some lattice L, [Ros86, §4] describes a construction of A^{∨} as a complex torus, which we now describe. Let

V^{∗} = \{f ∈ Hom_{R}(V, C) : f(αt) = \overline{α}f(t), all α ∈ C, t ∈ V \}.

Then V^{∗} is a complex vector space of the same dimension as V and the map ⟨f, v⟩ = Im f(t) is an R-linear pairing V^{∗} × V → R. Let

L^{∗} = {f ∈ V^{∗} : ⟨f, λ⟩ ∈ Z, all λ ∈ L}.

Since A is an abelian variety, there is a nondegenerate Riemann form H on A. The map λ : V → V^{∗} defined by λ(v) = H(v, ·) is an isomorphism of complex vector spaces. If v ∈ L, then λ(v) = H(v, ·) is integer valued on L, so λ(L) ⊂ L^{∗}. Thus λ induces an isogeny of complex tori V/L → V^{∗}/L^{∗}, so by Lemma 13.3.7 the torus V^{∗}/L^{∗} possesses a nondegenerate Riemann form (it’s a multiple of H). In [Ros86, §4], Rosen describes an explicit isomorphism between V^{∗}/L^{∗} and A^{∨}(C).

13.4.4 The Néron-Severi group and polarizations

Let A be an abelian variety over a field k. Recall that Pic(A) is the group of invertible sheaves on A, and Pic^{0}(A) is the subgroup of invertible sheaves that are algebraically equivalent to 0. The Néron-Severi group of A is the quotient Pic(A)/ Pic^{0}(A), so by definition we have an exact sequence

0 → Pic^{0}(A) → Pic(A) → NS(A) → 0.

Suppose L is an invertible sheaf on A. One can show that the map A(k) → Pic^{0}(A) defined by a ↦ t_{a}^{∗}L ⊗ L^{−1} is induced by homomorphism φ_{L} : A → A^{∨}. (Here t_{a}^{∗}L is the pullback of the sheaf L by translation by a.) Moreover, the map L ↦ φ_{L} induces a homomorphism from Pic(A) → Hom(A, A^{∨}) with kernel Pic^{0}(A). The group Hom(A, A^{∨}) is free of finite rank, so NS(A) is a free abelian group of finite rank. Thus Pic^{0}(A) is saturated in Pic(A) (i.e., the cokernel of the inclusion Pic^{0}(A) → Pic(A) is torsion free).

Definition 13.4.1 (Polarization). A polarization on A is a homomorphism λ : A → A^{∨} such that λ\overline{k} = φ_{L} for some L ∈ Pic(A\overline{k}). A polarization is principal if it is an isomorphism.

When the base field k is algebraically closed, the polarizations are in bijection with the elements of NS(A). For example, when dim A = 1, we have NS(A) = Z, and the polarizations on A are multiplication by n, for each integer n.

13.4.5 The Dual is functorial

The association A ↦ A^{∨} extends to a contravariant functor on the category of abelian varieties. Thus if φ : A → B is a homomorphism, there is a natural choice of homomorphism φ^{∨} : B^{∨} → A^{∨}. Also, (A^{∨})^{∨} = A and (φ^{∨})^{∨} = φ.

Theorem 13.4.2 below describes the kernel of φ^{∨} in terms of the kernel of φ. If G is a finite group scheme, the Cartier dual of G is Hom(G, G_{m}). For example, the Cartier dual of Z/mZ is μ_{m} and the Cartier dual of μ_{m} is Z/mZ. (If k is algebraically closed, then the Cartier dual of G is just G again.)

Theorem 13.4.2. If φ : A → B is a surjective homomorphism of abelian varieties with kernel G, so we have an exact sequence 0 → G → A → B → 0, then the kernel of φ^{∨} is the Cartier dual of G, so we have an exact sequence 0 → G^{∨} → B^{∨} → A^{∨} → 0.

13.5 Jacobians of curves

We begin this lecture about Jacobians with an inspiring quote of David Mumford:

“The Jacobian has always been a corner-stone in the analysis of algebraic curves and compact Riemann surfaces. [...] Weil’s construction [of the Jacobian] was the basis of his epoch-making proof of the Riemann Hypothesis for curves over finite fields, which really put characteristic p algebraic geometry on its feet.” – Mumford, Curves and Their Jacobians, page 49.

13.5.1 Divisors on curves and linear equivalence

Let X be a projective nonsingular algebraic curve over an algebraically field k. A divisor on X is a formal finite Z-linear combination ∑_{i=1}^{m} n_{i}P_{i} of closed points in X. Let Div(X) be the group of all divisors on X. The degree of a divisor ∑_{i=1}^{m} n_{i}P_{i} is the integer ∑_{i=1}^{m} n_{i}. Let Div^{0}(X) denote the subgroup of divisors of degree 0.

Suppose k is a perfect field (for example, k has characteristic 0 or k is finite), but do not require that k be algebraically closed. Let the group of divisors on X over k be the subgroup

Div(X) = Div(X/k) = H^{0}(Gal(\overline{k}/k), Div(X/\overline{k}))

of elements of Div(X/\overline{k}) that are fixed by every automorphism of \overline{k}/k. Likewise, let Div^{0}(X/k) be the elements of Div(X/k) of degree 0.

A rational function on an algebraic curve X is a function X → P^{1}, defined by polynomials, which has only a finite number of poles. For example, if X is the elliptic curve over k defined by y^{2} = x^{3} +ax+b, then the field of rational functions on X is the fraction field of the integral domain k[x, y]/(y^{2} − (x^{3} + ax + b)). Let K(X) denote the field of all rational functions on X defined over k.

There is a natural homomorphism K(X)^{∗} → Div(X) that associates to a rational function f its divisor

(f) = ∑ ord_{P} (f) · P

where ord_{P} (f) is the order of vanishing of f at P. Since X is nonsingular, the local ring of X at a point P is isomorphic to k[[t]]. Thus we can write f = t^{r}g(t) for some unit g(t) ∈ k[[t]]. Then R = ord_{P} (f). Example 13.5.1. If X = P^{1}, then the function f = x has divisor (0) − (∞). If X is the elliptic curve defined by y^{2} = x^{3} + ax + b, then

(x) = (0, \sqrt{b}) + (0, −\sqrt{b}) − 2∞,

and

(y) = (x_{1}, 0) + (x_{2}, 0) + (x_{3}, 0) − 3∞,

where x_{1}, x_{2}, and x_{3} are the roots of x^{3}+ax+b = 0. A uniformizing parameter t at the point ∞ is x/y. An equation for the elliptic curve in an affine neighborhood of ∞ is Z = X^{3} + aXZ^{2} + bZ^{3} (where ∞ = (0, 0) with respect to these coordinates) and x/y = X in these new coordinates. By repeatedly substituting Z into this equation we see that Z can be written in terms of X.

It is a standard fact in the theory of algebraic curves that if f is a nonzero rational function, then (f) ∈ Div^{0}(X), i.e., the number of poles of f equals the number of zeros of f. For example, if X is the Riemann sphere and f is a polynomial, then the number of zeros of f (counted with multiplicity) equals the degree of f, which equals the order of the pole of f at infinity.

The Picard group Pic(X) of X is the group of divisors on X modulo linear equivalence. Since divisors of functions have degree 0, the subgroup Pic^{0}(X) of divisors on X of degree 0, modulo linear equivalence, is well defined. Moreover, we have an exact sequence of abelian groups

0 → K(X)^{∗} → Div^{0}(X) → Pic^{0}(X) → 0.

Thus for any algebraic curve X we have associated to it an abelian group Pic^{0}(X). Suppose π : X → Y is a morphism of algebraic curves. If D is a divisor on Y, the pullback π^{∗}(D) is a divisor on X, which is defined as follows. If P ∈ Div(Y/\overline{k}) is a point, let π^{∗}(P) be the sum ∑ e_{Q/P} Q where π(Q) = P and e_{Q/P} is the ramification degree of Q/P. (Remark: If t is a uniformizer at P then e_{Q/P} = ord_{Q}(π^{∗}t_{P}).) One can show that π^{∗} : Div(Y) → Div(X) induces a homomorphism Pic^{0}(Y) → Pic^{0}(X). Furthermore, we obtain the contravariant Picard functor from the category of algebraic curves over a fixed base field to the category of abelian groups, which sends X to Pic^{0}(X) and π : X → Y to π^{∗} : Pic^{0}(Y) → Pic^{0}(X).

Alternatively, instead of defining morphisms by pullback of divisors, we could consider the push forward. Suppose π : X → Y is a morphism of algebraic curves and D is a divisor on X. If P ∈ Div(X/\overline{k}) is a point, let π_{∗}(P) = π(P). Then π_{∗} induces a morphism Pic^{0}(X) → Pic^{0}(Y). We again obtain a functor, called the covariant Albanese functor from the category of algebraic curves to the category of abelian groups, which sends X to Pic^{0}(X) and π : X → Y to π_{∗} : Pic^{0}(X) → Pic^{0}(Y).

13.5.2 Algebraic definition of the Jacobian

First we describe some universal properties of the Jacobian under the hypothesis that X(k) ≠ ∅. Thus suppose X is an algebraic curve over a field k and that X(k) ≠ ∅. The Jacobian variety of X is an abelian variety J such that for an extension k′/k, there is a (functorial) isomorphism J(k′) → Pic^{0}(X/k′). (I don’t know whether this condition uniquely characterizes the Jacobian.)

Fix a point P ∈ X(k). Then we obtain a map f : X(k) → Pic^{0}(X/k) by sending Q ∈ X(k) to the divisor class of Q−P. One can show that this map is induced by an injective morphism of algebraic varieties X ↪ J. This morphism has the following universal property: if A is an abelian variety and g : X → A is a morphism that sends P to 0 ∈ A, then there is a unique homomorphism ψ : J → A of abelian varieties such that g = ψ ∘ f:

\xymatrix{X \ar[r]^{f} \ar[dr]_{g} & J \ar[d]^{ψ} \\  & A}

This condition uniquely characterizes J, since if f′ : X → J′ and J′ has the universal property, then there are unique maps J → J′ and J′ → J whose composition in both directions must be the identity (use the universal property with A = J and f = g).

If X is an arbitrary curve over an arbitrary field, the Jacobian is an abelian variety that represents the “sheafification” of the “relative Picard functor”. Look in Milne’s article or Bosch-Lüktebohmert-Raynaud Neron Models for more details. Knowing this totally general definition won’t be important for this course, since we will only consider Jacobians of modular curves, and these curves always have a rational point, so the above properties will be sufficient.

A useful property of Jacobians is that they are canonically principally polarized, by a polarization that arises from the “θ divisor” on J. In particular, there is always an isomorphism J → J^{∨} = Pic^{0}(J).

13.5.3 The Abel-Jacobi theorem

Over the complex numbers, the construction of the Jacobian is classical. It was first considered in the 19th century in order to obtain relations between integrals of rational functions over algebraic curves (see Mumford’s book, Curves and Their Jacobians, Ch. III, for a nice discussion).

Let X be a Riemann surface, so X is a one-dimensional complex manifold. Thus there is a system of coordinate charts (U_{α}, t_{α}), where t_{α} : U_{α} → C is a homeomorphism of U_{α} onto an open subset of C, such that the change of coordinate maps are analytic isomorphisms. A differential 1-form on X is a choice of two continuous functions f and g to each local coordinate z = x + iy on U_{α} ⊂ X such that f dx + g dy is invariant under change of coordinates (i.e., if another local coordinate patch U′_{α} intersects U_{α}, then the differential is unchanged by the change of coordinate map on the overlap). If γ : [0, 1] → X is a path and ω = f dx + g dy is a 1-form, then

∫_{γ} ω := ∫_{0}^{1} (f(x(t), y(t))\frac{dx}{dt} + g(x(t), y(t))\frac{dy}{dt} ) dt ∈ C.

From complex analysis one sees that if γ is homologous to γ′, then ∫_{γ} ω = ∫_{γ′} ω. In fact, there is a nondegenerate pairing

H^{0}(X, Ω_{X}^{1}) × H_{1}(X, Z) → C

If X has genus g, then it is a standard fact that the complex vector space H^{0}(X, Ω_{X}^{1}) of holomorphic differentials on X is of dimension g. The integration pairing defined above induces a homomorphism from integral homology to the dual V of the differentials:

Φ : H_{1}(X, Z) → V = Hom(H^{0}(X, Ω_{X}^{1}), C).

This homomorphism is called the period mapping.

Theorem 13.5.2 (Abel-Jacobi). The image of Φ is a lattice in V.

The proof involves repeated clever application of the residue theorem.

The intersection pairing

H_{1}(X, Z) × H_{1}(X, Z) → Z

defines a nondegenerate alternating pairing on L = Φ(H_{1}(X, Z)). This pairing satisfies the conditions to induce a nondegenerate Riemann form on V, which gives J = V/L to structure of abelian variety. The abelian variety J is the Jacobian of X, and if P ∈ X, then the functional ω ↦ ∫_{P}^{Q} ω defines an embedding of X into J. Also, since the intersection pairing is perfect, it induces an isomorphism from J to J^{∨}. Example 13.5.3. For example, suppose X = X_{0}(23) is the modular curve attached to the subgroup Γ_{0}(23) of matrices in SL_{2}(Z) that are upper triangular modulo 24. Then g = 2, and a basis for H_{1}(X_{0}(23), Z) in terms of modular symbols is

{−1/19, 0}, {−1/17, 0}, {−1/15, 0}, {−1/11, 0}.

The matrix for the intersection pairing on this basis is

(\begin{matrix}0 & −1 & −1 & −1 \\ 1 & 0 & −1 & −1 \\ 1 & 1 & 0 & −1 \\ 1 & 1 & 1 & 0 \end{matrix})

With respect to a reduced integral basis for

H^{0}(X, Ω_{X}^{1}) ≅ S_{2}(Γ_{0}(23)),

the lattice Φ(H_{1}(X, Z)) of periods is (approximately) spanned by

[
(0.59153223605591049412844857432 - 1.68745927346801253993135357636*i
   0.762806324458047168681080323846571478727 - 0.60368764497868211035115379488*i),
(-0.59153223605591049412844857432 - 1.68745927346801253993135357636*i
   -0.762806324458047168681080323846571478727 - 0.60368764497868211035115379488*i),
(-1.354338560513957662809528899804 - 1.0837716284893304295801997808568748714097*i
   -0.59153223605591049412844857401 + 0.480083983510648319229045987467*i),
(-1.52561264891609433736216065099 0.342548176804273349105263499648)
]

13.5.4 Every abelian variety is a quotient of a Jacobian

Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety. The modular abelian varieties that we will encounter later are, by definition, exactly the quotients of the Jacobian J_{1}(N) of X_{1}(N) for some N. In this section we see that merely being a quotient of a Jacobian does not endow an abelian variety with any special properties.

Theorem 13.5.4 (Matsusaka). Let A be an abelian variety over an algebraically closed field. Then there is a Jacobian J and a surjective map J → A.

This was originally proved in On a generating curve of an abelian variety, Nat. Sc. Rep. Ochanomizu Univ. 3 (1952), 1–4. Here is the Math Review by P. Samuel:

An abelian variety A is said to be generated by a variety V (and a mapping f of V into A) if A is the group generated by f(V). It is proved that every abelian variety A may be generated by a curve defined over the algebraic closure of def(A). A first lemma shows that, if a variety V is the carrier of an algebraic system (X(M))_{M∈U} of curves (X(M) being defined, non-singular and disjoint from the singular bunch of V for almost all M in the parametrizing variety U) if this system has a simple base point on V, and if a mapping f of V into an abelian variety is constant on some X(M_{0}), then f is a constant; this is proved by specializing on M_{0} a generic point M of U and by using specializations of cycles [Matsusaka, Mem. Coll. Sci. Kyoto Univ. Ser. A. Math. 26, 167–173 (1951); these Rev. 13, 379]. Another lemma notices that, for a normal projective variety V, a suitable linear family of plane sections of V may be taken as a family (X(M)). Then the main result follows from the complete reducibility theorem. This result is said to be the basic tool for generalizing Chow’s theorem (”the Jacobian variety of a curve defined over k is an abelian projective variety defined over k”).

Milne [Mil86, §10] proves the theorem under the weaker hypothesis that the base field is infinite. We briefly sketch his proof now. If dim A = 1, then A is the Jacobian of itself, so we may assume dim A > 1. Embed A into P^{n}, then, using the Bertini theorem, cut A ⊂ P^{n} by hyperplane sections dim(A) − 1 times to obtain a nonsingular curve C on A of the form A∩V, where V is a linear subspace of P^{n}. Using standard arguments from Hartshorne [Har77], Milne shows (Lemma 10.3) that if W is a nonsingular variety and π : W → A is a finite morphism, then π^{−1}(C) is geometrically connected (the main point is that the pullback of an ample invertible sheaf by a finite morphism is ample). (A morphism f : X → Y is finite if for every open affine subset U = Spec(R) ⊂ Y, the inverse image f^{−1}(U) ⊂ X is an affine open subset Spec(B) with B a finitely generated R-module. Finite morphisms have finite fibers, but not conversely.) We assume this lemma and deduce the theorem.

Let J be the Jacobian of C; by the universal property of Jacobians there is a unique homomorphism f : J → A coming from the inclusion C ↪ A. The image A_{1} = f(J) is an abelian subvariety since images of homomorphisms of abelian varieties are abelian varieties. By the Poincare reducibility theorem (we only proved this over C, but it is true in general), there is an abelian subvariety A_{2} ⊂ A such that A_{1} + A_{2} = A and A_{1} ∩ A_{2} is finite. The isogeny g : A_{1} × A_{2} → A given by g(x, y) = x + y ∈ A is a finite morphism (any isogeny of abelian varieties is finite, flat, and surjective by Section 8 of [Mil86]). The inverse image g^{−1}(A_{1}) is a union of #(A_{1} ∩ A_{2}) irreducible components; if this intersection is nontrivial, then likewise g^{−1}(C) is reducible, which is a contradiction. This does not complete the proof, since it is possible that g is an isomorphism, so we use one additional trick. Suppose n is a positive integer coprime to the residue characteristic, and let

h = 1 × [n] : A_{1} × A_{2} → A_{1} × A_{2}

be the identity map on the first factor and multiplication by n on the second. Then h is finite and (h ∘ g)^{−1}(A_{1}) is a union of n^{2 dim A_{2}} = deg(h) irreducible components, hence (h ∘ g)^{−1}(C) is reducible, a contradiction.

Question 13.5.5. Is Theorem 13.5.4 false for some abelian variety A over some finite field k?

Question 13.5.6 (Milne). Using the theorem we can obtain a sequence of Jacobian varieties J_{1}, J_{2}, … that form a complex

⋯ → J_{2} → J_{1} → A → 0.

(In each case the image of J_{i+1} is the connected component of the kernel of J_{i} → J_{i−1}.) Is it possible to make this construction in such a way that the sequence terminates in 0?

Question 13.5.7 (Yau). Let A be an abelian variety. What can be said about the minimum of the dimensions of all Jacobians J such that there is a surjective morphism J → A?

Remark 13.5.8. Brian Conrad has explained to the author that if A is an abelian variety over an infinite field, then A can be embedded in a Jacobian J. This does not follow directly from Theorem 13.5.4 above, since if J → → A^{∨}, then the dual map A → J need not be injective.

13.6 Néron models

The main references for Néron models are as follows:

1. [AEC2]: Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Chapter IV of this book contains an extremely well written and motivated discussion of Néron models of elliptic curves over Dedekind domains with perfect residue field. In particular, Silverman gives an almost complete construction of Néron models of elliptic curves. Silverman very clearly really wants his reader to understand the construction. Highly recommended.

2. [BLR]: Bosch, Lütkebohmert, Raynaud, Néron Models. This is an excellent and accessible book that contains a complete construction of Néron models and some of their generalizations, a discussion of their functorial properties, and a sketch of the construction of Jacobians of families of curves. The goal of this book was to redo in scheme-theoretic language Néron original paper, which is written in a language that was ill-adapted to the subtleties of Néron models.

3. Artin, Néron Models, in Cornell-Silverman. This is the first-ever exposition of Néron’s original paper in the language of schemes.

13.6.1 What are Néron models?

Suppose E is an elliptic curve over Q. If Δ is the minimal discriminant of E, then E has good reduction at p for all p ∤ Δ, in the sense that E extends to an abelian scheme E over Z_{p} (i.e., a “smooth” and “proper” group scheme). One can not ask for E to extend to an abelian scheme over Z_{p} for all p | Δ. One can, however, ask whether there is a notion of “good” model for E at these bad primes. To quote [BLR, page 1], “It came as a surprise for arithmeticians and algebraic geometers when A. Néron, relaxing the condition of properness and concentrating on the group structure and the smoothness, discovered in the years 1961–1963 that such models exist in a canonical way.”

Before formally defining Néron models, we describe what it means for a morphism f : X → Y of schemes to be smooth. A morphism f : X → Y is finite type if for every open affine U = Spec(R) ⊂ Y there is a finite covering of f^{−1}(U) by open affines Spec(S), such that each S is a finitely generated R-algebra.

Definition 13.6.1. A morphism f : X → Y is smooth at x ∈ X if it is of finite type and there are open affine neighborhoods Spec(A) ⊂ X of x and Spec(R) ⊂ Y of f(x) such that

A ≅ R[t_{1}, …, t_{n+r}]/(f_{1}, …, f_{n})

for elements f_{1}, …, f_{n} ∈ R[t_{1}, …, t_{n+r}] and all n × n minors of the Jacobian matrix (∂f_{i}/∂t_{j}) generate the unit ideal of A. The morphism f is étale at x if, in addition, r = 0. A morphism is smooth of relative dimension d if it is smooth at x for every x ∈ X and r = d in the isomorphism above.

Smooth morphisms behave well. For example, if f and g are smooth and f ∘ g is defined, then f ∘ g is automatically smooth. Also, smooth morphisms are closed under base extension: if f : X → Y is a smooth morphism over S, and S′ is a scheme over S, then the induced map X×_{S}S′ → Y ×_{S}S′ is smooth. (If you’ve never seen products of schemes, it might be helpful to know that Spec(A) × Spec(B) = Spec(A⊗B). Read [Har77, §II.3] for more information about fiber products, which provide a geometric way to think about tensor products. Also, we often write X_{S′} as shorthand for X ×_{S} S′.)

We are now ready for the definition. Suppose R is a Dedekind domain with field of fractions K (e.g., R = Z and K = Q).

Definition 13.6.2 (Néron model). Let A be an abelian variety over K. The Néron model A of A is a smooth commutative group scheme over R such that for any smooth morphism S → R the natural map of abelian groups

Hom_{R}(S, A) → Hom_{K}(S ×_{R} K, A)

is a bijection. This is called the Néron mapping property: In more compact notation, it says that there is an isomorphism A(S) ≅ A(S_{K}).

Taking S = A in the definition we see that A is unique, up to a unique isomorphism.

It is a deep theorem that Néron models exist. Fortunately, Bosch, Lütkebohmert, and Raynaud devoted much time to create a carefully written book [BLR90] that explains the construction in modern language. Also, in the case of elliptic curves, Silverman’s second book [Sil94] is extremely helpful.

The basic idea of the construction is to first observe that if we can construct a Néron model at each localization R_{𝔭} at a nonzero prime ideal of R, then each of these local models can be glued to obtain a global Néron model (this uses that there are only finitely many primes of bad reduction). Thus we may assume that R is a discrete valuation ring.

The next step is to pass to the “strict henselization” R′ of R. A local ring R with maximal ideal ℘ is henselian if “every simple root lifts uniquely”; more precisely, if whenever f(x) ∈ R[x] and α ∈ R is such that f(α) ≡ 0 (mod ℘) and f′(α) ≢ 0 (mod ℘), there is a unique element α̃ ∈ R such that α̃ ≡ α (mod ℘) and f(α̃) = 0. The strict henselization of a discrete valuation ring R is an extension of R that is henselian and for which the residue field of R′ is the separable closure of the residue field of R (when the residue field is finite, the separable close is just the algebraic closure). The strict henselization is not too much bigger than R, though it is typically not finitely generated over R. It is, however, much smaller than the completion of R (e.g., Z_{p} is uncountable). The main geometric property of a strictly henselian ring R with residue field k is that if X is a smooth scheme over R, then the reduction map X(R) → X(k) is surjective.

Working over the strict henselization, we first resolve singularities. Then we use a generalization of the theorem that Weil used to construct Jacobians to pass from a birational group law to an actual group law. We thus obtain the Néron model over the strict henselization of R. Finally, we use Grothendieck’s faithfully flat descent to obtain a Néron model over R.

When A is the Jacobian of a curve X, there is an alternative approach that involves the “minimal proper regular model” of X. For example, when A is an elliptic curve, it is the Jacobian of itself, and the Néron model can be constructed in terms of the minimal proper regular model X of A as follows. In general, the model X → R is not also smooth. Let X′ be the smooth locus of X → R, which is obtained by removing from each closed fiber X_{F_{p}} = ∑ n_{i}C_{i} all irreducible components with multiplicity n_{i} ≥ 2 and all singular points on each C_{i}, and all points where at least two C_{i} intersect each other. Then the group structure on A extends to a group structure on X′, and X′ equipped with this group structure is the Néron model of A.

Explicit determination of the possibilities for the minimal proper regular model of an elliptic curve was carried out by Kodaira, then Néron, and finally in a very explicit form by Tate. Tate codified a way to find the model in what’s called “Tate’s Algorithm” (see Antwerp IV, which is available on my web page:

http://modular.fas.harvard.edu/scans/antwerp/, and look at Silverman, chap-

ter IV, which also has important implementation advice).

13.6.2 The Birch and Swinnerton-Dyer conjecture and Néron models

Throughout this section, let A be an abelian variety over Q and let A be the corresponding Néron model over Z. We work over Q for simplicity, but could work over any number field.

Let L(A, s) be the Hasse-Weil L-function of A (see Section [to be written]^{1} ). 1 Let r = ord_{s=1} L(A, s) be the analytic rank of A. The Birch and Swinnerton-Dyer Conjecture asserts that A(Q) ≈ Z^{r} ⊕ A(Q)_{tor} and

\frac{L^{(r)}(A, 1)}{r!} = \frac{(∏ c_{p}) · Ω_{A} · Reg_{A}· #� (A)}{#A(Q)_{tor} · #A^{∨}(Q)_{tor}}.

We have not defined most of the quantities appearing in this formula. In this section, we will define the Tamagawa numbers c_{p}, the real volume Ω_{A}, and the Shafarevich-Tate group � (A) in terms of the Néron model A of A.

We first define the Tamagawa numbers c_{p}, which are the orders groups of connected components. Let p be a prime and consider the closed fiber A_{F_{p}}, which is a smooth commutative group scheme over F_{p}. Then A_{F_{p}} is a disjoint union of one or more connected components. The connected component A_{F_{p}}^{0} that contains the identity element is a subgroup of A_{F_{p}} (Intuition: the group law is continuous and the continuous image of a connected set is connected, so the group structure restricts to A_{F_{p}}^{0}).

Definition 13.6.3 (Component Group). The component group of A at p is

Φ_{A,p} = A_{F_{p}}/A_{F_{p}}^{0}.

Fact: The component group Φ_{A,p} is a finite flat group scheme over F_{p}, and^{2} for 2 all but finitely many primes p, we have Φ_{A,p} = 0.

Definition 13.6.4 (Tamagawa Numbers). The Tamagawa number of A at a prime p is

c_{p} = #Φ_{A,p}(F_{p}).

1Add reference.

2Reference?

Next we define the real volume Ω_{A}. Choose a basis

ω_{1}, …, ω_{d} ∈ H^{0}(A, Ω_{A/Z}^{1})

for the global differential 1-forms on A, where d = dim A. The wedge product w = ω_{1} ∧ω_{2} ∧⋯∧ω_{d} is a global d-form on A. Then w induces a differential d-form on the real Lie group A(R).

Definition 13.6.5 (Real Volume). The real volume of A is

Ω_{A} = |∫_{A(R)} w| ∈ R_{>0}.

Finally, we give a definition of the Shafarevich-Tate group in terms of the Néron model. Let A_{0} be the scheme obtained from the Néron model A over A by removing from each closed fiber all nonidentity components. Then A_{0} is again a smooth commutative group scheme, but it need not have the Néron mapping property.

Recall that an étale morphism is a morphism that is smooth of relative dimension 0. A sheaf of abelian groups on the étale site Z_{ét} is a functor (satisfying certain axioms) from the category of étale morphism X → Z to the category of abelian groups. There are enough sheaves on Z_{ét} so that there is a cohomology theory for such sheaves, which is called étale cohomology. In particular if F is a sheaf on Z_{ét}, then for every integer q there is an abelian group H^{q}(Z_{ét}, F) associated to F that has the standard properties of a cohomology functor.

The group schemes A_{0} and A both determine sheaves on the étale site, which we will also denote by A_{0} and A.

Definition 13.6.6 (Shafarevich-Tate Group). Suppose A(R) is connected, i.e., that A_{0} = A. Then the Shafarevich-Tate group of A is H^{1}(Z_{ét}, A). More generally, suppose only that A(R) is connected. Then the Shafarevich-Tate group is the image of the natural map

f : H^{1}(Z_{ét}, A_{0}) → H^{1}(Z_{ét}, A).

Even more generally, if A(R) is not connected, then there is a natural map r : H^{1}(Z_{ét}, A) → H^{1}(Gal(C/R), A(C)) and � (A) = im(f) ∩ ker(r).

Mazur proves in the appendix to [Maz72] that this definition is equivalent to the usual Galois cohomology definition. To do this, he considers the exact sequence 0 → A_{0} → A → Φ_{A} → 0, where Φ_{A} is a sheaf version of ⊕_{p}Φ_{A,p}. The main input is Lang’s Theorem, which implies that over a local field, unramified Galois cohomology is the same as the cohomology of the corresponding component group.^{3}

3

Conjecture 13.6.7 (Shafarevich-Tate). The group H^{1}(Z_{ét}, A) is finite.

When A has rank 0, all component groups Φ_{A,p} are trivial, A(R) is connected, and A(Q)_{tor} and A^{∨}(Q)_{tor} are trivial, the Birch and Swinnerton-Dyer conjecture takes the simple form

\frac{L(A, 1)}{Ω_{A}} = # H^{1}(Z_{ét}, A).

Later^{4} , when A is modular, we will (almost) interpret L(A, 1)/Ω_{A} as the order of 4 a certain group that involves modular symbols. Thus the BSD conjecture asserts that two groups have the same order; however, they are not isomorphic, since, e.g., when dim A = 1 the modular symbols group is always cyclic, but the Shafarevich-Tate group is never cyclic (unless it is trivial).

3Reference for Lang’s Lemma, etc.

4Where?

13.6.3 Functorial properties of Neron models

The definition of Néron model is functorial, so one might expect the formation of Néron models to have good functorial properties. Unfortunately, it doesn’t.

Proposition 13.6.8. Let A and B be abelian varieties. If A and B are the Néron models of A and B, respectively, then the Néron model of A × B is A × B.

Suppose R ⊂ R′ is a finite extension of discrete valuation rings with fields of fractions K ⊂ K′. Sometimes, given an abelian variety A over a field K, it is easier to understand properties of the abelian variety, such as reduction, over K′. For example, you might have extra information that implies that A_{K′} decomposes as a product of well-understood abelian varieties. It would thus be useful if the Néron model of A_{K′} were simply the base extension A_{R′} of the Néron model of A over R. This is, however, frequently not the case.

Distinguishing various types of ramification will be useful in explaining how Néron models behave with respect to base change, so we now recall the notions of tame and wild ramification. If π generates the maximal ideal of R and v′ is the valuation on R′, then the extension is unramified if v′(π) = 1. It is tamely ramified if v′(π) is not divisible by the residue characteristic of R, and it is wildly ramified if v′(π) is divisible by the residue characteristic of R. For example, the extension Q_{p}(p^{1/p}) of Q_{p} is wildly ramified.

Example 13.6.9. If R is the ring of integers of a p-adic field, then for every integer n there is a unique unramified extension of R of degree n. See [Cp86, §I.7], where Fröhlich uses Hensel’s lemma to show that the unramified extensions of K = Frac(R) are in bijection with the finite (separable) extensions of the residue class field.

The Néron model does not behave well with respect to base change, except in some special cases. For example, suppose A is an abelian variety over the field of fractions K of a discrete valuation ring R. If K′ is the field of fractions of a finite unramified extension R′ of R, then the Néron model of A_{K′} is A_{R′}, where A is the Néron model of A over R. Thus the Néron model over an unramified extension is obtained by base extending the Néron model over the base. This is not too surprising because in the construction of Néron model we first passed to the strict henselization of R, which is a limit of unramified extensions.

Continuing with the above notation, if K′ is tamely ramified over K, then in general A_{R′} need not be the Néron model of A_{K′}. Assume that K′ is Galois over K. In [Edi92a], Bas Edixhoven describes the Néron model of A_{K} in terms of A_{R′}. To describe his main theorem, we introduce the restriction of scalars of a scheme.

Definition 13.6.10 (Restriction of Scalars). Let S′ → S be a morphism of schemes and let X′ be a scheme over S′. Consider the functor

R(T) = Hom_{S′}(T ×_{S} S′, X′)

on the category of all schemes T over S. If this functor is representable, the representing object X = Res_{S′/S}(X′) is called the restriction of scalars of X′ to S.

Edixhoven’s main theorem is that if G is the Galois group of K′ over K and X = Res_{R′/R}(A_{R′}) is the restriction of scalars of A_{R′} down to R, then there is a natural map A → X whose image is the closed subscheme X^{G} of fixed elements.

We finish this section with some cautious remarks about exactness properties of Néron models. If 0 → A → B → C → 0 is an exact sequence of abelian varieties, then the functorial definition of Néron models produces a complex of Néron models

0 → A → B → C → 0,

where A, B, and C are the Néron models of A, B, and C, respectively. This complex can fail to be exact at every point. For an in-depth discussion of conditions when we have exactness, along with examples that violate exactness, see [BLR90, Ch. 7], which says: “we will see that, except for quite special cases, there will be a defect of exactness, the defect of right exactness being much more serious than the one of left exactness.”

To give examples in which right exactness fails, it suffices to give an optimal quotient B → C such that for some p the induced map Φ_{B,p} → Φ_{C,p} on component groups is not surjective (recall that optimal means A = ker(B → C) is an abelian variety). Such quotients, with B and C modular, arise naturally in the context of Ribet’s level optimization. For example, the elliptic curve E given by y^{2} + xy = x^{3} + x^{2} − 11x is the optimal new quotient of the Jacobian J_{0}(33) of X_{0}(33). The component group of E at 3 has order 6, since E has semistable reduction at 3 (since 3 || 33) and ord_{3}(j(E)) = −6. The image of the component group of J_{0}(33) in the component group of E has order 2:

> OrderOfImageOfComponentGroupOfJ0N(ModularSymbols("33A"),3);
2

Note that the modular form associated to E is congruent modulo 3 to the form corresponding to J_{0}(11), which illustrates the connection with level optimization.

Abelian Varieties Attached to Modular Forms

In this chapter we describe how to decompose J_{1}(N), up to isogeny, as a product of abelian subvarieties A_{f} corresponding to Galois conjugacy classes of cusp forms f of weight 2. This was first accomplished by Shimura (see [Shi94, Theorem 7.14]). We also discuss properties of the Galois representation attached to f. ^{1} 1

In this chapter we will work almost exclusively with J_{1}(N). However, everything goes through exactly as below with J_{1}(N) replaced by J_{0}(N) and S_{2}(Γ_{1}(N)) replaced by S_{2}(Γ_{0}(N)). Since, J_{1}(N) has dimension much larger than J_{0}(N), so for computational investigations it is frequently better to work with J_{0}(N).

See Brian Conrad’s appendix to [ribet-stein: Lectures on Serre’s Conjectures] for a much more extensive exposition of the construction discussed below, which is geared toward preparing the reader for Deligne’s more general construction of Galois representations associated to newforms of weight k ≥ 2 (for that, see Conrad’s book ...).

14.1 Decomposition of the Hecke algebra

Let N be a positive integer and let

T = Z[…, T_{n}, …] ⊂ End(J_{1}(N))

be the algebra of all Hecke operators acting on J_{1}(N). Recall from Section 9.4 that the anemic Hecke algebra is the subalgebra

T_{0} = Z[…, T_{n}, …: (n, N) = 1] ⊂ T

of T obtained by adjoining to Z only those Hecke operators T_{n} with n relatively prime to N.

1Rewrite intro after chapter is done, and point to where each thing is done.

Remark 14.1.1. Viewed as Z-modules, T_{0} need not be saturated in T, i.e., T/T_{0} need not be torsion free. For example, if T is the Hecke algebra associated to S_{2}(Γ_{1}(24)) then T/T_{0} ≅ Z/2Z. Also, if T is the Hecke algebra associated to S_{2}(Γ_{0}(54)), then T/T_{0} ≅ Z/3Z × Z. ^{2} 2

If f = ∑ a_{n}q^{n} is a newform, then the field K_{f} = Q(a_{1}, a_{2}, …) has finite degree over Q, since the a_{n} are the eigenvalues of a family of commuting operators with integral characteristic polynomials. The Galois conjugates of f are the newforms σ(f) = ∑ σ(a_{n})q^{n}, for σ ∈ Gal(\overline{Q}/Q). There are [K_{f} : Q] Galois conjugates of f.

As in Section 9.4, we have a canonical decomposition

T_{0} ⊗ Q ≅ ∏_{f} K_{f},  (14.1.1)

where f varies over a set of representatives for the Galois conjugacy classes of newforms in S_{2}(Γ_{1}(N)) of level dividing N. For each f, let

π_{f} = (0, …, 0, 1, 0, …, 0) ∈ ∏ K_{f}

be projection onto the factor K_{f} of the product (14.1.1). Since T_{0} ⊂ T, and T has no additive torsion, we have T_{0} ⊗ Q ⊂ T ⊗ Q, so these projectors π_{f} lie in T_{Q} = T ⊗ Q. Since T_{Q} is commutative and the π_{f} are mutually orthogonal idempotents whose sum is (1, 1, …, 1), we see that T_{Q} breaks up as a product of algebras

T_{Q} ≅ ∏_{f} L_{f}, t ↦ ∑_{f} tπ_{f}.

14.1.1 The Dimension of the algebras L_{f}

Proposition 14.1.2. If f, L_{f} and K_{f} are as above, then dim_{K_{f}} L_{f} is the number of divisors of N/N_{f} where N_{f} is the level of the newform f.

Proof. Let V_{f} be the complex vector space spanned by all images of Galois conjugates of f via all maps α_{d} with d | N/N_{f}. It follows from [Atkin-Lehner-Li theory – multiplicity one]^{3} that the images via α_{d} of the Galois conjugates 3 of f are linearly independent. (Details: More generally, if f and g are newforms of level M, then by Proposition 9.2.1, B(f) = {α_{d}(f) : d | N/N_{f}} is a linearly independent set and likewise for B(g). Suppose some nonzero element f′ of the span of B(f) equals some element g′ of the span of B(g). Since T_{p}, for p ∤ N, commutes with α_{d}, we have T_{p}(f′) = a_{p}(f)f′ and T_{p}(g′) = a_{p}(g)g′, so 0 = T_{p}(0) = T_{p}(f′ − g′) = a_{p}(f)f′ − a_{p}(g)g′. Since f′ = g′, this implies that a_{p}(f) = a_{p}(g). Because a newform is determined by the eigenvalues of T_{p} for p ∤ N, it follows that f = g.) Thus the C-dimension of V_{f} is the number of divisors of N/N_{f} times dim_{Q} K_{f}.

The factor L_{f} is isomorphic to the image of T_{Q} ⊂ End(S_{k}(Γ_{1}(N))) in End(V_{f}). As in Section ??, there is a single element v ∈ V_{f} so that V_{f} = T_{C} · v. Thus the image of T_{Q} in End(V_{f}) has dimension dim_{C} V_{f}, and the result follows.

2I’m including the MAGMA scripts I used to check this as comments in the latex source, until I find the right way to justify these computational remarks. Maybe the remarks should point to an appendix where they are all justified?

3fill in

14.2 Decomposition of J_{1}(N) 133

Let’s examine a particular case of this proposition. Suppose p is a prime and f = ∑ a_{n}q^{n} is a newform of level N_{f} coprime to p, and let N = p · N_{f}. We will show that

L_{f} = K_{f}[U]/(U^{2} − a_{p}U + p),  (14.1.2)

hence dim_{K_{f}} L_{f} = 2 which, as expected, is the number of divisors of N/N_{f} = p. The first step is to view L_{f} as the space of operators generated by the Hecke operators T_{n} acting on the span V of the images f(dz) = f(q^{d}) for d | (N/N_{f}) = p. If n ≠ p, then T_{n} acts on V as the scalar a_{n}, and when n = p, the Hecke operator T_{p} acts on S_{k}(Γ_{1}(p · N_{f})) as the operator also denoted U_{p}. By Section 9.2, we know that U_{p} corresponds to the matrix ( \begin{matrix}a_{p} & 1 \\ −p & 0 \end{matrix} ) with respect to the basis f(q), f(q^{p})

of V. Thus U_{p} satisfies the relation U_{p}^{2} − a_{p}U + p. Since U_{p} is not a scalar matrix, this minimal polynomial of U_{p} is quadratic, which proves (14.1.2).

More generally, see [DDT94, Lem. 4.4] (Diamond-Darmon-Taylor)^{4} for an ex- 4 plicit presentation of L_{f} as a quotient

L_{f} ≅ K_{f}[…, U_{p}, …]/I

where I is an ideal and the U_{p} correspond to the prime divisors of N/N_{f}.

14.2 Decomposition of J_{1}(N)

Let f be a newform in S_{2}(Γ_{1}(N)) of level a divisor M of N, so f ∈ S_{2}(Γ_{1}(M))_{new} is a normalized eigenform for all the Hecke operators of level M. We associate to f an abelian subvariety A_{f} of J_{1}(N), of dimension [L_{f} : Q], as follows. Recall that π_{f} is the fth projector in T_{0} ⊗ Q = ∏_{g} K_{g}. We can not define A_{f} to be the image of J_{1}(N) under π_{f}, since π_{f} is only, a priori, an element of End(J_{1}(N))⊗Q. Fortunately, there exists a positive integer n such that nπ_{f} ∈ End(J_{1}(N)), and we let

A_{f} = nπ_{f}(J_{1}(N)).

This is independent of the choice of n, since the choices for n are all multiples of the “denominator” n_{0} of π_{f}, and if A is any abelian variety and n is a positive integer, then nA = A.

The natural map ∏_{f} A_{f} → J_{1}(N), which is induced by summing the inclusion maps, is an isogeny. Also A_{f} is simple if f is of level N, and otherwise A_{f} is isogenous to a power of A′_{f} ⊂ J_{1}(N_{f}). Thus we obtain an isogeny decomposition of J_{1}(N) as a product of Q-simple abelian varieties. Remark 14.2.1. The abelian varieties A_{f} frequently decompose further over \overline{Q}, i.e., they are not absolutely simple, and it is an interesting problem to determine an isogeny decomposition of J_{1}(N)\overline{Q} as a product of simple abelian varieties. It is still not known precisely how to do this computationally for any particular N. ^{5} 5

This decomposition can be viewed in another way over the complex numbers. As a complex torus, J_{1}(N)(C) has the following model:

J_{1}(N)(C) = Hom(S_{2}(Γ_{1}(N)), C)/H_{1}(X_{1}(N), Z).

4Remove parens.

5Add more/pointers/etc.

The action of the Hecke algebra T on J_{1}(N)(C) is compatible with its action on the cotangent space S_{2}(Γ_{1}(N)). This construction presents J_{1}(N)(C) naturally as V/L with V a complex vector space and L a lattice in V. The anemic Hecke algebra T_{0} then decomposes V as a direct sum V = ⨁_{f} V_{f}. The Hecke operators act on V_{f} and L in a compatible way, so T_{0} decomposes L ⊗ Q in a compatible way. Thus L_{f} = V_{f} ∩ L is a lattice in V_{f}, so we may A_{f}(C) view as the complex torus V_{f}/L_{f}.

Lemma 14.2.2. Let f ∈ S_{2}(Γ_{1}(N)) be a newform of level dividing N and A_{f} = nπ_{f}(J_{1}(N)) be the corresponding abelian subvariety of J_{1}(N). Then the Hecke algebra T ⊂ End(J_{1}(N)) leaves A_{f} invariant.

Proof. The Hecke algebra T is commutative, so if t ∈ T, then

tA_{f} = tnπ_{f}(J_{1}(N)) = nπ_{f}(tJ_{1}(N)) ⊂ nπ_{f}(J_{1}(N)) = A_{f}.

Remark 14.2.3. Viewing A_{f}(C) as V_{f}/L_{f} is extremely useful computationally, since L can be computed using modular symbols, and L_{f} can be cut out using the Hecke operators. For example, if f and g are nonconjugate newforms of level dividing N, we can explicitly compute the group structure of A_{f} ∩ A_{g} ⊂ J_{1}(N) by doing a computation with modular symbols in L. More precisely, we have

A_{f} ∩ A_{g} ≅ (L/(L_{f} + L_{g}))_{tor}.

Note that A_{f} depends on viewing f as an element of S_{2}(Γ_{1}(N)) for some N. Thus it would be more accurate to denote A_{f} by A_{f,N}, where N is any multiple of the level of f, and to reserve the notation A_{f} for the case N = 1. Then dim A_{f,N} is dim A_{f} times the number of divisors of N/N_{f}.

14.2.1 Aside: intersections and congruences

Suppose f and g are not Galois conjugate. Then the intersection Ψ = A_{f} ∩ A_{g} is finite, since V_{f} ∩ V_{g} = 0, and the integer #Ψ is of interest. This cardinality is related to congruence between f and g, but the exact relation is unclear. For example, one might expect that p | #Ψ if and only if there is a prime ℘ of the compositum K_{f}.K_{g} of residue characteristic p such that a_{q}(f) ≡ a_{q}(g) (mod ℘) for all q ∤ N. If p | #Ψ, then such a prime ℘ exists (take ℘ to be induced by a maximal ideal in the support of the nonzero T-module Ψ[p]). The converse is frequently true, but is sometimes false. For example, if N is the prime 431 and

f = q − q^{2} + q^{3} − q^{4} + q^{5} − q^{6} − 2q^{7} + ⋯
g = q − q^{2} + 3q^{3} − q^{4} − 3q^{5} − 3q^{6} + 2q^{7} + ⋯,

then f ≡ g (mod 2), but A_{f} ∩ A_{g} = 0. This example implies that “multiplicity one fails” for level 431 and p = 2, so the Hecke algebra associated to J_{0}(431) is not Gorenstein (see [Lloyd Kilford paper]^{6} for more details). 6

6fix reference

14.3 Galois representations attached to A_{f} 135

14.3 Galois representations attached to A_{f}

It is important to emphasize the case when f is a newform of level N, since then A_{f} is Q-simple^{7} and there is a compatible family of 2-dimensional ℓ-adic 7 representations attached to f, which arise from torsion points on A_{f}.

Proposition 14.1.2 implies that L_{f} = K_{f}. Fix such an f, let A = A_{f}, let K = K_{f}, and let

d = dim A = dim_{Q} K = [K : Q].

Let ℓ be a prime and consider the Q_{ℓ}-adic Tate module Tate_{ℓ}(A) of A:

Tate_{ℓ}(A) = Q_{ℓ} ⊗ \underleftarrow{lim}_{ν>0} A[ℓ^{ν}].

Note that as a Q_{ℓ}-vector space Tate_{ℓ}(A) ≅ Q_{ℓ}^{2d}, since A[n] ≅ (Z/nZ)^{2d}, as groups.

There is a natural action of the ring K ⊗_{Q} Q_{ℓ} on Tate_{ℓ}(A). By algebraic number theory

K ⊗_{Q} Q_{ℓ} = ∏_{λ|ℓ} K_{λ},

where λ runs through the primes of the ring O_{K} of integers of K lying over ℓ and K_{λ} denotes the completion of K with respect to the absolute value induced by λ. Thus Tate_{ℓ}(A) decomposes as a product

Tate_{ℓ}(A) = ∏_{λ|ℓ} Tate_{λ}(A)

where Tate_{λ}(A) is a K_{λ} vector space.

Lemma 14.3.1. Let the notation be as above. Then for all λ lying over ℓ,

dim_{K_{λ}} Tate_{λ}(A) = 2.

Proof. Write A = V/L, with V = V_{f} a complex vector space and L a lattice. Then Tate_{λ}(A) ≅ L ⊗ Q_{ℓ} as K_{λ}-modules (not as Gal(\overline{Q}/Q)-modules!), since A[ℓ^{n}] ≅ L/ℓ^{n}L, and \underleftarrow{lim}_{n} L/ℓ^{n}L ≅ Z_{ℓ}⊗L. Also, L⊗Q is a vector space over K, which must have dimension 2, since L ⊗ Q has dimension 2d = 2 dim A and K has degree d. Thus

Tate_{λ}(A) ≅ L ⊗ K_{λ} ≈ (K ⊕ K) ⊗_{K} K_{λ} ≅ K_{λ} ⊕ K_{λ}

has dimension 2 over K_{λ}.

Now consider Tate_{λ}(A), which is a K_{λ}-vector space of dimension 2. The Hecke operators are defined over Q, so Gal(\overline{Q}/Q) acts on Tate_{ℓ}(A) in a way compatible with the action of K ⊗_{Q} Q_{ℓ}. We thus obtain a homomorphism

ρ_{ℓ} = ρ_{f,ℓ} : Gal(\overline{Q}/Q) → Aut_{K⊗Q_{ℓ}} Tate_{ℓ}(A) ≈ GL_{2}(K ⊗ Q_{ℓ}) ≅ ∏_{λ} GL_{2}(K_{λ}).

Thus ρ_{ℓ} is the direct sum of ℓ-adic Galois representations ρ_{λ} where

ρ_{λ} : Gal(\overline{Q}/Q) → End_{K_{λ}}(Tate_{λ}(A))

7add pointer to where this is proved.

gives the action of Gal(\overline{Q}/Q) on Tate_{λ}(A).

If p ∤ ℓN, then ρ_{λ} is unramified at p (see [ST68, Thm. 1]). In this case it makes sense to consider ρ_{λ}(φ_{p}), where φ_{p} ∈ Gal(\overline{Q}/Q) is a Frobenius element at p. Then ρ_{λ}(φ_{p}) has a well-defined trace and determinant, or equivalently, a well-defined characteristic polynomial Φ(X) ∈ K_{λ}[X].

Theorem 14.3.2. Let f ∈ S_{2}(Γ_{1}(N), ε) be a newform of level N with Dirichlet character ε. Suppose p ∤ ℓN, and let φ_{p} ∈ Gal(\overline{Q}/Q) be a Frobenius element at p. Let Φ(X) be the characteristic polynomial of ρ_{λ}(φ_{p}). Then

Φ(X) = X^{2} − a_{p}X + p · ε(p),

where a_{p} is the pth coefficient of the modular form f (thus a_{p} is the image of T_{p} in E_{f} and ε(p) is the image of ⟨p⟩).

Let φ = φ_{p}. By the Cayley-Hamilton theorem

ρ_{λ}(φ)^{2} − tr(ρ_{λ}(φ))ρ_{λ}(φ) + det(ρ_{λ}(φ)) = 0.

Using the Eichler-Shimura congruence relation (see^{8} ) we will show that tr(ρ_{λ}(φ)) = 8 a_{p}, but we defer the proof of this until ...^{9} . 9

We will prove that det(ρ_{λ}(φ)) = p in the special case when ε = 1. This will follow from the equality

det(ρ_{λ}) = χ_{ℓ},  (14.3.1)

where χ_{ℓ} is the ℓth cyclotomic character

χ_{ℓ} : Gal(\overline{Q}/Q) → Z_{ℓ}^{∗} ⊂ K_{λ}^{∗},

which gives the action of Gal(\overline{Q}/Q) on μ_{ℓ^{∞}}. We have χ_{ℓ}(φ) = p because φ induces induces pth powering map on μ_{ℓ^{∞}}.

It remains to establish (14.3.1). The simplest case is when A is an elliptic curve. In [Sil92, ]^{10} , Silverman shows that det(ρ_{ℓ}) = χ_{ℓ} using the Weil pairing. We will 10 consider the Weil pairing in more generality in the next section, and use it to establish (14.3.1).

14.3.1 The Weil pairing

Let T_{ℓ}(A) = \underleftarrow{lim}_{n≥1} A[ℓ^{n}], so Tate_{ℓ}(A) = Q_{ℓ} ⊗ T_{ℓ}(A). The Weil pairing is a nondegenerate perfect pairing

e_{ℓ} : T_{ℓ}(A) × T_{ℓ}(A^{∨}) → Z_{ℓ}(1).

(See e.g., [Mil86, §16] for a summary of some of its main properties.) Remark 14.3.3. Identify Z/ℓ^{n}Z with μ_{ℓ^{n}} by 1 ↦ e^{−2πi/ℓ^{n}}, and extend to a map Z_{ℓ} → Z_{ℓ}(1). If J = Jac(X) is a Jacobian, then the Weil pairing on J is induced by the canonical isomorphism

T_{ℓ}(J) ≅ H^{1}(X, Z_{ℓ}) = H^{1}(X, Z) ⊗ Z_{ℓ},

8Next week!

9when?

10get ref

and the cup product pairing

H^{1}(X, Z_{ℓ}) ⊗_{Z_{ℓ}} H^{1}(X, Z_{ℓ}) ⟶^{∪} Z_{ℓ}.

For more details see the discussion on pages 210–211 of Conrad’s appendix to [RS01], and the references therein. In particular, note that H^{1}(X, Z_{ℓ}) is isomorphic to H_{1}(X, Z_{ℓ}), because H_{1}(X, Z_{ℓ}) is self-dual because of the intersection pairing. It is easy to see that H_{1}(X, Z_{ℓ}) ≅ T_{ℓ}(J) since by Abel-Jacobi J ≅ T_{0}(J)/ H_{1}(X, Z), where T_{0}(J) is the tangent space at J at 0 (see Lemma 14.3.1). ^{11} 11

Here Z_{ℓ}(1) ≅ \underleftarrow{lim} μ_{ℓ^{n}} is isomorphic to Z_{ℓ} as a ring, but has the action of Gal(\overline{Q}/Q) induced by the action of Gal(\overline{Q}/Q) on \underleftarrow{lim} μ_{ℓ^{n}}. Given σ ∈ Gal(\overline{Q}/Q), there is an element χ_{ℓ}(σ) ∈ Z_{ℓ}^{∗} such that σ(ζ) = ζ^{χ_{ℓ}(σ)}, for every ℓ^{n}th root of unity ζ. If we view Z_{ℓ}(1) as just Z_{ℓ} with an action of Gal(\overline{Q}/Q), then the action of σ ∈ Gal(\overline{Q}/Q) on Z_{ℓ}(1) is left multiplication by χ_{ℓ}(σ) ∈ Z_{ℓ}^{∗}.

Definition 14.3.4 (Cyclotomic Character). The homomorphism

χ_{ℓ} : Gal(\overline{Q}/Q) → Z_{ℓ}^{∗}

is called the ℓ-adic cyclotomic character.

If φ : A → A^{∨} is a polarization (so it is an isogeny defined by translation of an ample invertible sheaf), we define a pairing

e_{ℓ}^{φ} : T_{ℓ}(A) × T_{ℓ}(A) → Z_{ℓ}(1)  (14.3.2)

by e_{ℓ}^{φ}(a, b) = e_{ℓ}(a, φ(b)). The pairing (14.3.2) is a skew-symmetric, nondegenerate, bilinear pairing that is Gal(\overline{Q}/Q)-equivariant, in the sense that if σ ∈ Gal(\overline{Q}/Q), then

e_{ℓ}^{φ}(σ(a), σ(b)) = σ · e_{ℓ}^{φ}(a, b) = χ_{ℓ}(σ)e_{ℓ}^{φ}(a, b).

We now apply the Weil pairing in the special case A = A_{f} ⊂ J_{1}(N). Abelian varieties attached to modular forms are equipped with a canonical polarization called the modular polarization. The canonical principal polarization of J_{1}(N) is an isomorphism J_{1}(N) ⟶^{∼} J_{1}(N)^{∨}, so we obtain the modular polarization φ = φ_{A} : A → A^{∨} of A, as illustrated in the following diagram:

[figure: J_{1}(N); ^{autoduality^{≅}}; J_{1}(N)^{∨}; A; polarization φ_{A}; A^{∨}]

Consider (14.3.2) with φ = φ_{A} the modular polarization. Tensoring over Q and restricting to Tate_{λ}(A), we obtain a nondegenerate skew-symmetric bilinear pairing

e : Tate_{λ}(A) × Tate_{λ}(A) → Q_{ℓ}(1).  (14.3.3)

The nondegeneracy follows from the nondegeneracy of e_{ℓ}^{φ} and the observation that

e_{ℓ}^{φ}(Tate_{λ}(A), Tate_{λ′}(A)) = 0

11Remove or expand?

when λ ≠ λ′. This uses the Galois equivariance of e_{ℓ}^{ϕ} carries over to Galois equivariance of e, in the following sense. If σ ∈ Gal(\overline{Q}/Q) and x, y ∈ Tate_{λ}(A), then

e(σx, σy) = σe(x, y) = χ_{ℓ}(σ)e(x, y).

Note that σ acts on Q_{ℓ}(1) as multiplication by χ_{ℓ}(σ).

14.3.2 The Determinant

There are two proofs of the theorem, a fancy proof and a concrete proof. We first present the fancy proof. The pairing e of (14.3.3) is a skew-symmetric and bilinear form so it determines a Gal(\overline{Q}/Q)-equivarient homomorphism

⋀_{K_{λ}}^{2} Tate_{λ}(A) → Q_{ℓ}(1).  (14.3.4)

It is not a priori true that we can take the wedge product over K_{λ} instead of Q_{ℓ}, but we can because e(tx, y) = e(x, ty) for any t ∈ K_{λ}. This is where we use that A is attached to a newform with trivial character, since when the character is nontrivial, the relation between e(T_{p}x, y) and e(x, T_{p}y) will involve ⟨p⟩. Let D = ⋀^{2} Tate_{λ}(A) and note that dim_{K_{λ}} D = 1, since Tate_{λ}(A) has dimension 2 over K_{λ}.

There is a canonical isomorphism

Hom_{Q_{ℓ}}(D, Q_{ℓ}(1)) ≅ Hom_{K_{λ}}(D, K_{λ}(1)),

and the map of (14.3.4) maps to an isomorphism D ≅ K_{λ}(1) of Gal(\overline{Q}/Q)-modules. Since the representation of Gal(\overline{Q}/Q) on D is the determinant, and the representation on K_{λ}(1) is the cyclotomic character χ_{ℓ}, it follows that det ρ_{λ} = χ_{ℓ}.

Next we consider a concrete proof. If σ ∈ Gal(\overline{Q}/Q), then we must show that det(σ) = χ_{ℓ}(σ). Choose a basis x, y ∈ Tate_{λ}(A) of Tate_{λ}(A) as a 2 dimensional K_{λ} vector space. We have σ(x) = ax + cy and σ(y) = bx + dy, for a, b, c, d ∈ K_{λ}. Then

χ_{ℓ}(σ)e(x, y) = ⟨σx, σy)
= e(ax + cy, bx + dy)
= e(ax, bx) + e(ax, dy) + e(cy, bx) + e(cy, dy)
= e(ax, dy) + e(cy, bx)
= e(adx, y) − e(bcx, y)
= e((ad − bc)x, y)
= (ad − bc)e(x, y)

To see that e(ax, bx) = 0, note that

e(ax, bx) = e(abx, x) = −e(x, abx) = −e(ax, bx).

Finally, since e is nondegenerate, there exists x, y such that e(x, y) ≠ 0, so χ_{ℓ}(σ) = ad − bc = det(σ).

14.4 Remarks about the modular polarization

Let A and φ be as in Section 14.3.1. The degree deg(φ) of the modular polarization of A is an interesting arithmetic invariant of A. If B ⊂ J_{1}(N) is the sum of all modular abelian varieties A_{g} attached to newforms g ∈ S_{2}(Γ_{1}(N)), with g not a Galois conjugate of f and of level dividing N, then ker(φ) ≅ A ∩ B, as illustrated in the following diagram:

\xymatrix{ & ker(φ_{B}) \ar[dr] \ar[d]^{≅} \\ ker(φ_{A}i) \ar[r]^{≅} \ar[dr] & A ∩ B \ar[r] \ar[d] & B \ar[d] \ar[dr] \\  & A \ar[r] \ar[dr]^{φ} & J_{1}(N) \ar[d] \ar[r] & B^{∨} \\  &  & A^{∨}}

Note that ker(φ_{B}) is also isomorphic to A ∩ B, as indicated in the diagram.

In connection with Section ??, the quantity ker(φ_{A}) = A∩B is closely related to congruences between f and eigenforms orthogonal to the Galois conjugates of f.

When A has dimension 1, we may alternatively view A as a quotient of X_{1}(N) via the map

X_{1}(N) → J_{1}(N) → A^{∨} ≅ A.

Then φ_{A} : A → A is pullback of divisors to X_{1}(N) followed by push forward, which is multiplication by the degree. Thus φ_{A} = [n], where n is the degree of the morphism X_{1}(N) → A of algebraic curves. The modular degree is

deg(X_{1}(N) → A) = \sqrt{deg(φ_{A})}.

More generally, if A has dimension greater than 1, then deg(φ_{A}) has order a perfect square (for references, see [Mil86, Thm. 13.3]), and we define the modular degree to be \sqrt{deg(φ_{A})}.

Let f be a newform of level N. In the spirit of Section 14.2.1 we use congruences to define a number related to the modular degree, called the congruence number. For a subspace V ⊂ S_{2}(Γ_{1}(N)), let V (Z) = V ∩Z[[q]] be the elements with integral q-expansion at ∞ and V^{⊥} denotes the orthogonal complement of V with respect to the Petersson inner product. The congruence number of f is

r_{f} = #\frac{S_{2}(Γ_{1}(N))(Z)}{V_{f}(Z) + V_{f}^{⊥}(Z)},

where V_{f} is the complex vector space spanned by the Galois conjugates of f. We thus have two positive associated to f, the congruence number r_{f} and the modular degree m_{f} of of A_{f}.

Theorem 14.4.1. m_{f} | r_{f}

Ribet mentions this in the case of elliptic curves in [ZAGIER, 1985] [Zag85a], but the statement is given incorrectly in that paper (the paper says that r_{f} | m_{f}, which is wrong). The proof for dimension greater than one is in [AGASHE-STEIN, Manin constant...]. Ribet also subsequently proved that if p^{2} ∤ N, then ord_{p}(m_{f}) = ord_{p}(r_{f}).

We can make the same definitions with J_{1}(N) replaced by J_{0}(N), so if f ∈ S_{2}(Γ_{0}(N)) is a newform, A_{f} ⊂ J_{0}(N), and the congruence number measures congruences between f and other forms in S_{2}(Γ_{0}(N)). In [FM99, Ques. 4.4], they ask whether it is always the case that m_{f} = r_{f} when A_{f} is an elliptic curve, and m_{f} and r_{f} are defined relative to Γ_{0}(N). I^{12} implemented an algorithm in MAGMA 12 to compute r_{f}, and found the first few counterexamples, which occur when

N = 54, 64, 72, 80, 88, 92, 96, 99, 108, 120, 124, 126, 128, 135, 144.

For example, the elliptic curve A labeled 54B1 in [Cre97] has r_{A} = 6 and m_{A} = 2. To see directly that 3 | r_{A}, observe that if f is the newform corresponding to E and g is the newform corresponding to X_{0}(27), then g(q) + g(q^{2}) is congruent to f modulo 3. This is consistent with Ribet’s theorem that if p | r_{A}/m_{A} then p^{2} | N. There seems to be no absolute bound on the p that occur.

It would be interesting to determine the answer to the analogue of the question of Frey-Mueller for Γ_{1}(N). For example, if A ⊂ J_{1}(54) is the curve isogeneous to 54B1, then m_{A} = 18 is divisible by 3. However, I do not know^{13} r_{A} in this case, 13 because I haven’t written a program to compute it for Γ_{1}(N).

12change...

13fix

Modularity of Abelian Varieties

15.1 Modularity over Q

Definition 15.1.1 (Modular Abelian Variety). Let A be an abelian variety over Q. Then A is modular if there exists a positive integer N and a surjective map J_{1}(N) → A defined over Q.

The following theorem is the culmination of a huge amount of work, which started with Wiles’s successful attack [Wil95] on Fermat’s Last Theorem, and finished with [BCDT01].

Theorem 15.1.2 (Breuil, Conrad, Diamond, Taylor, Wiles). Let E be an elliptic curve over Q. Then E is modular.

We will say nothing about the proof here.^{1} 1 If A is an abelian variety over Q, let End_{Q}(A) denote the ring of endomorphisms

of A that are defined over Q.

Definition 15.1.3 (GL_{2}-type). An abelian variety A over Q is of GL_{2}-type if the endomorphism algebra Q ⊗ End_{Q}(A) contains a number field of degree equal to the dimension of A.

For example, every elliptic curve E over Q is trivially of GL_{2}-type, since Q ⊂ Q ⊗ End_{Q}(E).

Proposition 15.1.4. If A is an abelian variety over Q, and K ⊂ Q ⊗ End_{Q}(A) is a field, then [K : Q] divides dim A.

Proof. As discussed in [Rib92, §2], K acts faithfully on the tangent space Tan_{0}(A/Q) over Q to A at 0, which is a Q vector space of dimension dim(A). Thus Tan_{0}(A/Q) is a vector space over K, hence has Q-dimension a multiple of [K : Q].

1Also pointer to later in book.

Proposition 15.1.4 implies, in particular, that if E is an elliptic curve over Q, then End_{Q}(E) = Q. Recall^{2} that E has CM or is a complex multiplication elliptic 2 curve if End\overline{Q}(E) ≠ Z). Proposition 15.1.4 implies that if E is a CM elliptic curve, the extra endomorphisms are never defined over Q.

Proposition 15.1.5. Suppose A = A_{f} ⊂ J_{1}(N) is an abelian variety attached to a newform of level N. Then A is of GL_{2}-type.

Proof. The endomorphism ring of A_{f} contains O_{f} = Z[…, a_{n}(f), …], hence the field K_{f} = Q(…, a_{n}(f), …) is contained in Q ⊗ End_{Q}(A). Since A_{f} = nπJ_{1}(N), where π is a projector onto the factor K_{f} of the anemic Hecke algebra T_{0} ⊗_{Z} Q, we have dim A_{f} = [K_{f} : Q]. (One way to see this is to recall that the tangent space T = Hom(S_{2}(Γ_{1}(N)), C) to J_{1}(N) at 0 is free of rank 1 over T_{0} ⊗_{Z} C.)

Conjecture 15.1.6 (Ribet). Every abelian variety over Q of GL_{2}-type is modular.

Suppose

ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{p})

is an odd irreducible continuous Galois representation, where odd means that

det(ρ(c)) = −1,

where c is complex conjugation. We say that ρ is modular if there is a newform f ∈ S_{k}(Γ_{1}(N)), and a prime ideal ℘ ⊂ O_{f} such that for all ℓ ∤ Np, we have

Tr(ρ(Frob_{ℓ})) ≡ a_{ℓ} (mod ℘),
Det(ρ(Frob_{ℓ})) ≡ ℓ^{k−1} · ε(ℓ) (mod ℘).

Here χ_{p} is the p-adic cyclotomic character, and ε is the (Nebentypus) character of the newform f.

Conjecture 15.1.7 (Serre). Every odd irreducible continuous representation

ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{p})

is modular. Moreover, there is a formula for the “optimal” weight k(ρ) and level N(ρ) of a newform that gives rise to ρ.

In [Ser87], Serre describes the formula for the weight and level. Also, it is now known due to work of Ribet, Edixhoven, Coleman, Voloch, Gross, and others that if ρ is modular, then ρ arises from a form of the conjectured weight and level, except in some cases when p = 2. (For more details see the survey paper [RS01].) However, the full Conjecture 15.1.7 is known in very few cases. Remark 15.1.8. There is interesting recent work of Richard Taylor which connects Conjecture 15.1.7 with the open question of whether every variety of a certain type has a point over a solvable extension of Q. The question of the existence of solvable points (“solvability of varieties in radicals”) seems very difficult. For example, we don’t even know the answer for genus one curves, or have a good reason to make a conjecture either way (as far as I know^{3} ). There’s a book of Mike Fried that 3 discusses this solvability question.^{4} 4

2from where

3fix

4Find the exact reference in that book and the exact book.

Serre’s conjecture is very strong. For example, it would imply modularity of all abelian varieties over Q that could possibly be modular, and the proof of this implication does not rely on Theorem 15.1.2.

Theorem 15.1.9 (Ribet). Serre’s conjectures on modularity of all odd irreducible mod p Galois representations implies Conjecture 15.1.6.

To give the reader a sense of the connection between Serre’s conjecture and modularity, we sketch some of the key ideas of the proof of Theorem 15.1.9; for more details the reader may consult Sections 1–4 of [Rib92].

Without loss, we may assume that A is Q-simple. As explained in the not trivial [Rib92, Thm. 2.1], this hypothesis implies that

K = Q ⊗_{Z} End_{Q}(A)

is a number field of degree dim(A). The Tate modules

Tate_{ℓ}(A) = Q_{ℓ} ⊗ \underleftarrow{lim}_{n≥1} A[ℓ^{n}]

are free of rank two over K ⊗ Q_{ℓ}, so the action of Gal(\overline{Q}/Q) on Tate_{ℓ}(A) defines a representation

ρ_{A,ℓ} : Gal(\overline{Q}/Q) → GL_{2}(K ⊗ Q_{ℓ}).

Remarks 15.1.10. That these representations take values in GL_{2} is why such A are said to be “of GL_{2}-type”. Also, note that the above applies to A = A_{f} ⊂ J_{1}(N), and the ℓ-adic representations attached to f are just the factors of ρ_{A,ℓ} coming from the fact that K ⊗ Q_{ℓ} ≅ ∏_{λ|ℓ} K_{λ}.

The deepest input to Ribet’s proof is Faltings’s isogeny theorem, which Faltings proved in order to prove Mordell’s conjecture (there are only a finite number of L-rational points on any curve over L of genus at least 2).

If B is an abelian variety over Q, let

L(B, s) = ∏_{all primes p} \frac{1}{det (1 − p^{−s} · Frob_{p} | Tate_{ℓ}(A))} = ∏_{p} L_{p}(B, s),

where ℓ is a prime of good reduction (it makes no difference which one).

Theorem 15.1.11 (Faltings). Let A and B be abelian varieties. Then A is isogenous to B if and only if L_{p}(A, s) = L_{p}(B, s) for almost all p.

Using an analysis of Galois representations and properties of conductors and applying results of Faltings, Ribet finds an infinite set Λ of primes of K such that all ρ_{A,λ} are irredudible and there only finitely many Serre invariants N(ρ_{A,λ}) and k(ρ_{A,λ}). For each of these λ, by Conjecture 15.1.7 there is a newform f_{λ} of level N(ρ_{A,λ})) and weight k(ρ_{A,λ}) that gives rise to the mod ℓ representation ρ_{A,λ}. Since Λ is infinite, but there are only finitely many Serre invariants N(ρ_{A,λ})), k(ρ_{A,λ}), there must be a single newform f and an infinite subset Λ′ of Λ so that for every λ ∈ Λ′ the newform f gives rise to ρ_{A,λ}.

Let B = A_{f} ⊂ J_{1}(N) be the abelian variety attached to f. Fix any prime p of good reduction. There are infinitely many primes λ ∈ Λ′ such that ρ_{A,λ} ≅ ρ_{B,λ̃} for some λ̃, and for these λ,

det (1 − p^{−s} · Frob_{p} |A[λ]) = det (1 − p^{−s} · Frob_{p} |B[λ̃]) .

This means that the degree two polynomials in p^{−s} (over the appropriate fields, e.g., K ⊗ Q_{ℓ} for A)

det (1 − p^{−s} · Frob_{p} | Tate_{ℓ}(A))

and

det (1 − p^{−s} · Frob_{p} | Tate_{ℓ}(B))

are congruent modulo infinitely many primes. Therefore they are equal. By Theorem 15.1.11, it follows that A is isogenous to B = A_{f}, so A is modular.

15.2 Modularity of elliptic curves over \overline{Q}

Definition 15.2.1 (Modular Elliptic Curve). An elliptic curve E over \overline{Q} is modular if there is a surjective morphism X_{1}(N) → E for some N.

Definition 15.2.2 (Q-curve). An elliptic curve E over Q-bar is a Q-curve if for every σ ∈ Gal(\overline{Q}/Q) there is an isogeny E^{σ} → E (over \overline{Q}).

Theorem 15.2.3 (Ribet). Let E be an elliptic curve over \overline{Q}. If E is modular, then E is a Q-curve, or E has CM.

This theorem is proved in [Rib92, §5].

Conjecture 15.2.4 (Ribet). Let E be an elliptic curve over \overline{Q}. If E is a Q-curve, then E is modular.

In [Rib92, §6], Ribet proves that Conjecture 15.1.7 implies Conjecture 15.2.4. He does this by showing that if a Q-curve E does not have CM then there is a Q-simple abelian variety A over Q of GL_{2}-type such that E is a simple factor of A over \overline{Q}. This is accomplished finding a model for E over a Galois extension K of Q, restricting scalars down to Q to obtain an abelian variety B = Res_{K/Q}(E), and using Galois cohomology computations (mainly in H^{2}’s) to find the required A of GL_{2}- type inside B. Then Theorem 15.1.9 and our assumption that Conjecture 15.1.7 is true together immediately imply that A is modular.

Ellenberg and Skinner [ES00] have recently used methods similar to those used by Wiles to prove strong theorems toward Conjecture 15.2.4. See also Ellenberg’s survey [Ell02], which discusses earlier modularity results of Hasegawa, Hashimoto, Hida, Momose, and Shimura, and gives an example to show that there are infinitely many Q-curves whose modularity is not known.

Theorem 15.2.5 (Ellenberg, Skinner). Let E be a Q-curve over a number field K with semistable reduction at all primes of K lying over 3, and suppose that K is unramified at 3. Then E is modular.

15.3 Modularity of abelian varieties over \overline{Q}

Hida discusses modularity of abelian varieties over \overline{Q} in [Hid00]. Let A be an abelian variety over \overline{Q}. For any subalgebra E ⊂ Q⊗End(A/\overline{Q}), let Q⊗End_{E}(A/\overline{Q}) be the subalgebra of endomorphism that commute with E.

Definition 15.3.1 (Real Multiplication Abelian Variety). An abelian variety A over \overline{Q} is a real multiplication abelian variety if there is a totally real field K with [K : Q] = dim(A) such that

K ⊂ Q ⊗ End(A/\overline{Q})  and  K = Q ⊗ End_{K}(A/\overline{Q}).

If E is a CM elliptic curve, then E is not a real multiplication abelian variety, because the extra CM endomorphisms in End(E/\overline{Q}) commute with K = Q, so K ≠ Q ⊗ End_{K}(A/\overline{Q}).

In analogy with Q-curves, Hida makes the following definition.

Definition 15.3.2 (Q-RMAV). Let A be an abelian variety over \overline{Q} with real multiplication by a field K. Then A is a Q-real multiplication abelian variety (abbreviated Q-RMAV) if for every σ ∈ Gal(\overline{Q}/Q) there is a K-linear isogeny A^{σ} → A. Here K acts on A^{σ} via the canonical isomorphism End(A) → End(A^{σ}), which exists since applying σ is nothing but relabeling everything.

I haven’t had sufficient time to absorb Hida’s paper to see just what he proves about such abelian varieties, so I’m not quite sure how to formulate the correct modularity conjectures and theorems in this generality.

Elizabeth Pyle’s Ph.D. thesis [Pyl] under Ribet about “building blocks” is also very relevant to this section.

146 15. Modularity of Abelian Varieties

16^{L-functions}

16.1 L-functions attached to modular forms

Let f = ∑_{n≥1} a_{n}q^{n} ∈ S_{k}(Γ_{1}(N)) be a cusp form. Definition 16.1.1 (L-series). The L-series of f is

L(f, s) = ∑_{n≥1} \frac{a_{n}}{n^{s}} .

Definition 16.1.2 (Λ-function). The completed Λ function of f is

Λ(f, s) = N^{s/2}(2π)^{−s}Γ(s)L(f, s),

where

Γ(s) = ∫_{0}^{∞} e^{−t}t^{s} \frac{dt}{t}

is the Γ function (so Γ(n) = (n − 1)! for positive integers n).

We can view Λ(f, s) as a (Mellin) transform of f, in the following sense: Proposition 16.1.3. We have

Λ(f, s) = N^{s/2} ∫_{0}^{∞} f(iy)y^{s} \frac{dy}{y} ,

and this integral converges for Re(s) > \frac{k}{2} + 1.

148 16. L-functions

Proof. We have

∫_{0}^{∞} f(iy)y^{s} \frac{dy}{y} = ∫_{0}^{∞} ∑_{n=1}^{∞} a_{n}e^{−2πny}y^{s} \frac{dy}{y}

= ∑_{n=1}^{∞} a_{n} ∫_{0}^{∞} e^{−t}(2πn)^{−s}t^{s} \frac{dt}{t} (t = 2πny)

= (2π)^{−s}Γ(s) ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

To go from the first line to the second line, we reverse the summation and integration and perform the change of variables t = 2πny. (We omit discussion of convergence.^{1} ) 1

16.1.1 Analytic continuation and functional equations

We define the Atkin-Lehner operator W_{N} on S_{k}(Γ_{1}(N)) as follows. If w_{N} = ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ), then [w_{N}^{2}]_{k} acts as (−N)^{k−2}, so if

W_{N}(f) = N^{1− \frac{k}{2}} · f|[w_{N}]_{k},

then W_{N}^{2} = (−1)^{k}. (Note that W_{N} is an involution when k is even.) It is easy to check directly that if γ ∈ Γ_{1}(N), then w_{N}γw_{N}^{−1} ∈ Γ_{1}(N), so W_{N} preserves S_{k}(Γ_{1}(N)). Note that in general W_{N} does not commute with the Hecke operators T_{p}, for p | N.

The following theorem is mainly due to Hecke (and maybe other people, at least in this generality). For a very general version of this theorem, see [Li75].^{2} 2

Theorem 16.1.4. Suppose f ∈ S_{k}(Γ_{1}(N), χ) is a cusp form with character χ. Then Λ(f, s) extends to an entire (holomorphic on all of C) function which satisfies the functional equation

Λ(f, s) = i^{k}Λ(W_{N}(f), k − s).

Since N^{s/2}(2π)^{−s}Γ(s) is everywhere nonzero, Theorem 16.1.4 implies that L(f, s) also extends to an entire function.

It follows from Definition 16.1.2 that Λ(cf, s) = cΛ(f, s) for any c ∈ C. Thus if f is a W_{N}-eigenform, so that W_{N}(f) = wf for some w ∈ C, then the functional equation becomes

Λ(f, s) = i^{k}wΛ(f, k − s).

If k = 2, then W_{N} is an involution, so w = ±1, and the sign in the functional equation is ε(f) = i^{k}w = −w, which is the negative of the sign of the Atkin-Lehner involution W_{N} on f. It is straightforward to show that ε(f) = 1 if and only if ord_{s=1} L(f, s) is even. Parity observations such as this are extremely useful when trying to understand the Birch and Swinnerton-Dyer conjecture.

1change for book

2Add refs. – see page 58 of Diamond-Im.

16.1 L-functions attached to modular forms 149

Sketch of proof of Theorem 16.1.4 when N = 1. We follow [Kna92, §VIII.5] closely.

Note that since w_{1} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ SL_{2}(Z), the condition W_{1}(f) = f is satisfied for any f ∈ S_{k}(1). This translates into the equality

f (−\frac{1}{z}) = z^{k}f(z).  (16.1.1)

Write z = x + iy with x and y real. Then (16.1.1) along the positive imaginary axis (so z = iy with y positive real) is

f (\frac{i}{y} ) = i^{k}y^{k}f(iy).  (16.1.2)

From Proposition 16.1.3 we have

Λ(f, s) = ∫_{0}^{∞} f(iy)y^{s−1}dy,  (16.1.3)

and this integral converges for Re(s) > \frac{k}{2} + 1.

Again using growth estimates, one shows that

∫_{1}^{∞} f(iy)y^{s−1}dy

converges for all s ∈ C, and defines an entire function. Breaking the path in (16.1.3) at 1, we have for Re(s) > \frac{k}{2} + 1 that

Λ(f, s) = ∫_{0}^{1} f(iy)y^{s−1}dy + ∫_{1}^{∞} f(iy)y^{s−1}dy.

Apply the change of variables t = 1/y to the first term and use (16.1.2) to get

∫_{0}^{1} f(iy)y^{s−1}dy = ∫_{∞}^{1} −f(i/t)t^{1−s} \frac{1}{t^{2}} dt
= ∫_{1}^{∞} f(i/t)t^{−1−s}dt
= ∫_{1}^{∞} i^{k}t^{k}f(it)t^{−1−s}dt
= i^{k} ∫_{1}^{∞} f(it)t^{k−1−s}dt.

Thus

Λ(f, s) = i^{k} ∫_{1}^{∞} f(it)t^{k−s−1}dt + ∫_{1}^{∞} f(iy)y^{s−1}dy.

The first term is just a translation of the second, so the first term extends to an entire function as well. Thus Λ(f, s) extends to an entire function.

The proof of the general case for Γ_{0}(N) is almost the same, except the path is broken at 1/\sqrt{N}, since i/\sqrt{N} is a fixed point for w_{N}.

16.1.2 A Conjecture about nonvanishing of L(f, k/2)

Suppose f ∈ S_{k}(1) is an eigenform. If k ≡ 2 (mod 4), then L(f, k/2) = 0 for reasons related to the discussion after the statement of Theorem 16.1.4. On the other hand, if k ≡ 0 (mod 4), then ord_{s=k/2} L(f, k/2) is even, so L(f, k/2) may or may not vanish.

Conjecture 16.1.5. Suppose k ≡ 0 (mod 4). Then L(f, k/2) ≠ 0.

According to [CF99], Conjecture 16.1.5 is true for weight k if there is some n such that the characteristic polynomial of T_{n} on S_{k}(1) is irreducible. Thus Maeda’s conjecture implies Conjecture 16.1.5. Put another way, if you find an f of level 1 and weight k ≡ 0 (mod 4) such that L(f, k/2) = 0, then Maeda’s conjecture is false for weight k.

16.1.3 Euler products

Euler products make very clear how L-functions of eigenforms encode deep arithmetic information about representations of Gal(\overline{Q}/Q). Given a “compatible family” of ℓ-adic representations ρ of Gal(\overline{Q}/Q), one can define an Euler product L(ρ, s), but in general it is very hard to say anything about the analytic properties of L(ρ, s). However, as we saw above, when ρ is attached to a modular form, we know that L(ρ, s) is entire.

Theorem 16.1.6. Let f = ∑ a_{n}q^{n} be a newform in S_{k}(Γ_{1}(N), ε), and let L(f, s) = ∑_{n≥1} a_{n}n^{−s} be the associated Dirichlet series. Then L(f, s) has an Euler product

L(f, s) = ∏_{p|N} \frac{1}{1 − a_{p}p^{−s}} · ∏_{p∤N} \frac{1}{1 − a_{p}p^{−s} + ε(p)p^{k−1}p^{−2s}} .

Note that it is not really necessary to separate out the factors with p | N as we have done, since ε(p) = 0 whenever p | N. Also, note that the denominators are of the form F (p^{−s}), where

F (X) = 1 − a_{p}X + ε(p)p^{k−1}X^{2}

is the reverse of the characteristic polynomial of Frob_{p} acting on any of the ℓ-adic representations attached to f, with p ≠ ℓ.

Recall that if p is a prime, then for every r ≥ 2 the Hecke operators satisfy the relationship

T_{p^{r}} = T_{p^{r−1}}T_{p} − p^{k−1}ε(p)T_{p^{r−2}}.  (16.1.4)

Lemma 16.1.7. For every prime p we have the formal equality

∑_{r≥0} T_{p^{r}}X^{r} = \frac{1}{1 − T_{p}X + ε(p)p^{k−1}X^{2}} .  (16.1.5)

Proof. Multiply both sides of (16.1.5) by 1 − T_{p}X + ε(p)p^{k−1}X^{2} to obtain the equation

∑_{r≥0} T_{p^{r}}X^{r} − ∑_{r≥0}(T_{p^{r}}T_{p})X^{r+1} + ∑_{r≥0}(ε(p)p^{k−1}T_{p^{r}})X^{r+2} = 1.

[[THERE IS A COMMENTED OUT POSTSCRIPT PICTURE – look at source and redo it in Sage]]

FIGURE 16.1.1. Graph of L(E, s) for s real, for curves of ranks 0 to 3.

This equation is true if and only if the lemma is true. Equality follows by checking the first few terms and shifting the index down by 1 for the second sum and down by 2 for the third sum, then using (16.1.4).

Note that ε(p) = 0 when p | N, so when p | N

∑_{r≥0} T_{p^{r}}X^{r} = \frac{1}{1 − T_{p}X} .

Since the eigenvalues a_{n} of f also satisfy (16.1.4), we obtain each factor of the Euler product of Theorem 16.1.6 by substituting the a_{n} for the T_{n} and p^{−s} for X into (16.1.4). For (n, m) = 1, we have a_{nm} = a_{n}a_{m}, so

∑_{n≥1} \frac{a_{n}}{n^{s}} = ∏_{p} (∑_{r≥0} \frac{a_{p^{r}}}{p^{rs}} ) ,

which gives the full Euler product for L(f, s) = ∑ a_{n}n^{−s}.

16.1.4 Visualizing L-function

A. Shwayder did his Harvard junior project with me^{3} on visualizing L-functions of 3 elliptic curves (or equivalently, of newforms f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) with a_{n} ∈ Z for all n. The graphs in Figures 16.1.1–?? of L(E, s), for s real, and |L(E, s)|, for s complex, are from his paper.

3refine

152 16. L-functions

The Birch and Swinnerton-Dyer Conjecture

This chapter is about the conjecture of Birch and Swinnerton-Dyer on the arithmetic of abelian varieties. We focus primarily on abelian varieties attached to modular forms.

In the 1960s, Sir Peter Swinnerton-Dyer worked with the EDSAC computer lab at Cambridge University, and developed an operating system that ran on that computer (so he told me once). He and Bryan Birch programmed EDSAC to compute various quantities associated to elliptic curves. They then formulated the conjectures in this chapter in the case of dimension 1 (see [Bir65, Bir71, SD67]). Tate formulated the conjectures in a functorial way for abelian varieties of arbitrary dimension over global fields in [Tat66], and proved that if the conjecture is true for an abelian variety A, then it is also true for each abelian variety isogenous to A.

Suitably interpreted, the conjectures may by viewed as generalizing the analytic class number formula, and Bloch and Kato generalized the conjectures to Grothendieck motives in [BK90].

17.1 The Rank conjecture

Let A be an abelian variety over a number field K.

Definition 17.1.1 (Mordell-Weil Group). The Mordell-Weil group of A is the abelian group AK) of all K-rational points on A.

Theorem 17.1.2 (Mordell-Weil). The Mordell-Weil group A(K) of A is finitely generated.

The proof is nontrivial and combines two ideas. First, one proves the “weak Mordell-Weil theorem”: for any integer m the quotient A(K)/mA(K) is finite. This is proved by combining Galois cohomology techniques with standard finiteness theorems from algebraic number theory. The second idea is to introduce the Néron-Tate canonical height h : A(K) → R_{≥0} and use properties of h to deduce, from finiteness of A(K)/mA(K), that A(K) itself is finitely generated.

Definition 17.1.3 (Rank). By the structure theorem A(K) ≅ Z^{r} ⊕ G_{tor}, where r is a nonnegative integer and G_{tor} is the torsion subgroup of G. The rank of A is r.

Let f ∈ S_{2}(Γ_{1}(N)) be a newform of level N, and let A = A_{f} ⊂ J_{1}(N) be the corresponding abelian variety. Let f_{1}, …, f_{d} denote the Gal(\overline{Q}/Q)-conjugates of f, so if f = ∑ a_{n}q^{n}, then f_{i} = ∑ σ(a_{n})q^{n}, for some σ ∈ Gal(\overline{Q}/Q).

Definition 17.1.4 (L-function of A). We define the L-function of A = A_{f} (or any abelian variety isogenous to A) to be

L(A, s) = ∏_{i=1}^{d} L(f_{i}, s).

By Theorem 16.1.4, each L(f_{i}, s) is an entire function on C, so L(A, s) is entire. In Section 17.4 we will discuss an intrinsic way to define L(A, s) that does not require that A be attached to a modular form. However, in general we do not know that L(A, s) is entire.

Conjecture 17.1.5 (Birch and Swinnerton-Dyer). The rank of A(Q) is equal to ord_{s=1} L(A, s).

One motivation for Conjecture 17.1.5 is the following formal observation. Assume for simplicity of notation that dim A = 1. By Theorem 16.1.6, the L-function L(A, s) = L(f, s) has an Euler product representation

L(A, s) = ∏_{p|N} \frac{1}{1 − a_{p}p^{−s}} · ∏_{p∤N} \frac{1}{1 − a_{p}p^{−s} + p · p^{−2s}} ,

which is valid for Re(s) sufficiently large. (Note that ε = 1, since A is a modular elliptic curve, hence a quotient of X_{0}(N).) There is no loss in considering the product L^{∗}(A, s) over only the good primes p ∤ N, since ord_{s=1} L(A, s) = ord_{s=1} L^{∗}(A, s) (because ∏_{p|N} \frac{1}{1−a_{p}p^{−s}} is nonzero at s = 1). We then have formally that

L^{∗}(A, 1) = ∏_{p∤N} \frac{1}{1 − a_{p}p^{−1} + p^{−1}}

= ∏_{p∤N} \frac{p}{p − a_{p} + 1}

= ∏_{p∤N} \frac{p}{#A(F_{p})}

The intuition is that if the rank of A is large, i.e., A(Q) is large, then each group A(F_{p}) will also be large since it has many points coming from reducing the elements of A(Q) modulo p. It seems likely that if the groups #A(F_{p}) are unusually large, then L^{∗}(A, 1) = 0, and computational evidence suggests the more precise Conjecture 17.1.5. Example 17.1.6. Let A_{0} be the elliptic curve y^{2} + y = x^{3} − x^{2}, which has rank 0 and conductor 11, let A_{1} be the elliptic curve y^{2} +y = x^{3} −x, which has rank 1 and conductor 37, let A_{2} be the elliptic curve y^{2} + y = x^{3} + x^{2} − 2x, which has rank 2 and conductor 389, and finally let A_{3} be the elliptic curve y^{2} + y = x^{3} − 7x + 6, which has rank 3 and conductor 5077. By an exhaustive search, these are known to be the smallest-conductor elliptic curves of each rank. Conjecture 17.1.5 is known to be true for them, the most difficult being A_{3}, which relies on the results of [GZ86].

The following diagram illustrates |#A_{i}(F_{p})| for p < 100, for each of these curves. The height of the red line (first) above the prime p is |#A_{0}(F_{p})|, the green line (second) gives the value for A_{1}, the blue line (third) for A_{2}, and the black line (fourth) for A_{3}. The intuition described above suggests that the clumps should look like triangles, with the first line shorter than the second, the second shorter than the third, and the third shorter than the fourth—however, this is visibly not the case. The large Mordell-Weil group over Q does not increase the size of every E(F_{p}) as much as we might at first suspect. Nonetheless, the first line is no longer than the last line for every p except p = 41, 79, 83, 97.

[[THERE IS A COMMENTED OUT POSTSCRIPT PICTURE – look at source and redo it in Sage]] Remark 17.1.7. Suppose that L(A, 1) ≠ 0. Then assuming the Riemann hypothesis for L(A, s) (i.e., that L(A, s) ≠ 0 for Re(s) > 1), Goldfeld [Gol82] proved that the Euler product for L(A, s), formally evaluated at 1, converges but does not converge to L(A, 1). Instead, it converges (very slowly) to L(A, 1)/\sqrt{2}. For further details and insight into this strange behavior, see [Con03]. Remark 17.1.8. The Clay Math Institute has offered a one million dollar prize for a proof of Conjecture 17.1.5 for elliptic curves over Q. See [Wil00].

Theorem 17.1.9 (Kolyvagin-Logachev). Suppose f ∈ S_{2}(Γ_{0}(N)) is a newform such that ord_{s=1} L(f, s) ≤ 1. Then Conjecture 17.1.5 is true for A_{f}.

Theorem 17.1.10 (Kato). Suppose f ∈ S_{2}(Γ_{1}(N)) and L(f, 1) ≠ 0. Then Conjecture 17.1.5 is true for A_{f}.

17.2 Refined rank zero conjecture

Let f ∈ S_{2}(Γ_{1}(N)) be a newform of level N, and let A_{f} ⊂ J_{1}(N) be the corresponding abelian variety.

The following conjecture refines Conjecture 17.1.5 in the case L(A, 1) ≠ 0. We recall some of the notation below, where we give a formula for L(A, 1)/Ω_{A}, which can be computed up to an vinteger, which we call the Manin index. Note that the definitions, results, and proofs in this section are all true exactly as stated with X_{1}(N) replaced by X_{0}(N), which is relevant if one wants to do computations.

Conjecture 17.2.1 (Birch and Swinnerton-Dyer). Suppose L(A, 1) ≠ 0. Then

\frac{L(A, 1)}{Ω_{A}} = \frac{#� (A) · ∏_{p|N} c_{p}}{#A(Q)_{tor} · #A^{∨}(Q)_{tor}} .

By Theorem 17.1.10, the group � (A) is finite, so the right hand side makes sense. The right hand side is a rational number, so if Conjecture 17.2.1 is true, then the quotient L(A, 1)/Ω_{A} should also be a rational number. In fact, this is [[THERE IS A COMMENTED OUT POSTSCRIPT PICTURE – look at source

and redo it in Sage]]

FIGURE 17.2.1. Graphs of real solutions to y^{2}z = x^{3} − xz^{2} on three affine patches

true, as we will prove below (see Theorem 17.2.11). Below we will discuss aspects of the proof of rationality in the case that A is an elliptic curve, and at the end of this section we give a proof of the general case.

In to more easily understanding L(A, 1)/Ω_{A}, it will be easiest to work with A = A_{f}^{∨}, where A_{f}^{∨} is the dual of A_{f}. We view A naturally as a quotient of J_{1}(N) as follows. Dualizing the map A_{f} ↪ J_{1}(N) we obtain a surjective map J_{1}(N) → A_{f}^{∨}. Passing to the dual doesn’t affect whether or not L(A, 1)/Ω_{A} is rational, since changing A by an isogeny does not change L(A, 1), and only changes Ω_{A} by multiplication by a nonzero rational number.

17.2.1 The Number of real components

Definition 17.2.2 (Real Components). Let c_{∞} be the number of connected components of A(R).

If A is an elliptic curve, then c_{∞} = 1 or 2, depending on whether the graph of the affine part of A(R) in the plane R^{2} is connected. For example, Figure 17.2.1 shows the real points of the elliptic curve defined by y^{2} = x^{3} − x in the three affine patches that cover P^{2}. The completed curve has two real components.

In general, there is a simple formula for c_{∞} in terms of the action of complex conjugation on H_{1}(A(R), Z), which can be computed using modular symbols. The formula is

log_{2}(c_{∞}) = dim_{F_{2}} A(R)[2] − dim(A).

17.2.2 The Manin index

The map J_{1}(N) → A induces a map J → A on Néron models. Pullback of differentials defines a map

H^{0}(A, Ω_{A/Z}^{1}) → H^{0}(J , Ω_{J /Z}^{1}).  (17.2.1)

One can show^{1} that there is a q-expansion map 1

H^{0}(J , Ω_{J /Z}^{1}) → Z[[q]]  (17.2.2)

which agrees with the usual q-expansion map after tensoring with C. (For us X_{1}(N) is the curve that parameterizes pairs (E, μ_{N} ↪ E), so that there is a q-expansion map with values in Z[[q]].)

Let φ_{A} be the composition of (17.2.1) with (17.2.2).

Definition 17.2.3 (Manin Index). The Manin index c_{A} of A is the index of φ_{A}(H^{0}(A, Ω_{A/Z}^{1})) in its saturation. I.e., it is the order of the quotient group

(\frac{Z[[q]]}{φ_{A}(H^{0}(A, Ω_{A/Z}^{1}))})_{tor} .

1reference or further discussion

Open Problem 17.2.4. Find an algorithm to compute c_{A}.

Manin conjectured that c_{A} = 1 when dim A = 1, and I think c_{A} = 1 in general.

Conjecture 17.2.5 (Agashe, Stein). c_{A} = 1.

This conjecture is false if A is not required to be attached to a newform, even if A_{f} ⊂ J_{1}(N)^{new}. For example, Adam Joyce, a student of Kevin Buzzard, found an A ⊂ J_{1}(431) (and also A′ ⊂ J_{0}(431)) whose Manin constant is 2. Here A is isogenous over Q to a product of two elliptic curves. ^{2} Also, the Manin index for 2 J_{0}(33) (viewed as a quotient of J_{0}(33)) is divisible by 3, because there is a cusp form in S_{2}(Γ_{0}(33)) that has integer Fourier expansion at ∞, but not at one of the other cusps.

Theorem 17.2.6. If f ∈ S_{2}(Γ_{0}(N)) then the Manin index c of A_{f}^{∨} can only divisible by 2 or primes whose square divides N. Moreover, if 4 ∤ N, then ord_{2}(c) ≤ dim(A_{f}).

The proof involves applying nontrivial theorems of Raynaud about exactness of sequences of differentials, then using a trick with the Atkin-Lehner involution, which was introduced by Mazur in [Maz78], and finally one applies the “q-expansion principle” in characteristic p to deduce the result (see [?]). Also, Edixhoven claims he can prove that if A_{f} is an elliptic curve then c_{A} is only divisible by 2, 3, 5, or 7. His argument use his semistable models for X_{0}(p^{2}), but my understanding is that the details are not all written up.^{3} 3

17.2.3 The Real volume Ω_{A}

Definition 17.2.7 (Real Volume). The real volume Ω_{A} of A(R) is the volume of A(R) with respect to a measure obtained by wedging together a basis for H^{0}(A, Ω^{1}).

If A is an elliptic curve with minimal Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

then one can show that

ω = \frac{dx}{2y + a_{1}x + a_{3}}  (17.2.3)

is a basis for H^{0}(A, Ω^{1}). Thus

Ω_{A} = ∫_{A(R)} \frac{dx}{2y + a_{1}x + a_{3}} .

There is a fast algorithm for computing Ω_{A}, for A an elliptic curve, which relies on the quickly-convergent Gauss arithmetic-geometric mean (see [Cre97, §3.7]). For example, if A is the curve defined by y^{2} = x^{3} − x (this is a minimal model), then

Ω_{A} ∼ 2 × 2.622057554292119810464839589.

For a general abelian variety A, it is an open problem to compute Ω_{A}. However, we can compute Ω_{A}/c_{A}, where c_{A} is the Manin index of A, by explicitly computing A as a complex torus using the period mapping Φ, which we define in the next section.

2Add better reference.

3Refine.

17.2.4 The Period mapping

Let

Φ : H_{1}(X_{1}(N), Z) → Hom_{C}(Cf_{1} + ⋯ + Cf_{d}, C)

be the period mapping on integral homology induced by integrating homology classes on X_{0}(N) against the C-vector space spanned by the Gal(\overline{Q}/Q)-conjugates f_{i} of f. Extend Φ to H_{1}(X_{1}(N), Q) by Q-linearity. We normalize Φ so that Φ({0, ∞})(f) = L(f, 1). More explicitly, for α, β ∈ P^{1}(Q), we have

Φ({α, β})(f) = −2πi ∫_{α}^{β} f(z)dz.

The motivation for this normalization is that

L(f, 1) = −2πi ∫_{0}^{i∞} f(z)dz,  (17.2.4)

which we see immediately from the Mellin transform definition of L(f, s):

L(f, s) = (2π)^{s}Γ(s)^{−1} ∫_{0}^{i∞}(−iz)^{s}f(z)\frac{dz}{z} .

17.2.5 The Manin-Drinfeld theorem

Recall the Manin-Drinfeld theorem, which we proved long ago, asserts that {0, ∞} ∈ H_{1}(X_{0}(N), Q). We proved this by explicitly computing (p + 1 − T_{p})({0, ∞}), for p ∤ N, noting that the result is in H_{1}(X_{0}(N), Z), and inverting p + 1 − T_{p}. Thus there is an integer n such that n{0, ∞} ∈ H_{1}(X_{0}(N), Z).

Suppose that A = A_{f}^{∨} is an elliptic curve quotient of J_{0}(N). Rewriting (17.2.4) in terms of Φ, we have Φ({0, ∞}) = L(f, 1). Let ω be a minimal differential on A, as in (17.2.3), so ω = −c_{A} · 2πif(z)dz, where c_{A} is the Manin index of A, and the equality is after pulling ω back to H^{0}(X_{0}(N), Ω) ≅ S_{2}(Γ_{0}(N)). Note that when we defined c_{A}, there was no factor of 2πi, since we compared ω with f(q)\frac{dq}{q}, and q = e^{2πiz}, so dq/q = 2πidz.

17.2.6 The Period lattice

The period lattice of A with respect to a nonzero differential g on A is

L_{g} = {∫_{γ} g : γ ∈ H_{1}(A, Z)} ,

and we have A(C) ≅ C/L_{g}. This is the Abel-Jacobi theorem, and the significance of g is that we are choosing a basis for the one-dimensional C-vector space Hom(H^{0}(A, Ω), C), in order to embed the image of H_{1}(A, Z) in C.

The integral ∫_{A(R)} g is “visible” in terms of the complex torus representation of A(C) = C/L_{g}. More precisely, if L_{g} is not rectangular, then A(R) may be identified with the part of the real line in a fundamental domain for L_{g}, and ∫_{A(R)} g is the length of this segment of the real line. If L_{g} is rectangular, then it is that line along with another line above it that is midway to the top of the fundamental domain.

The real volume, which appears in Conjecture 17.2.1, is

Ω_{A} = ∫_{A(R)} ω = −c_{A} · 2πi ∫_{A(R)} f.

Thus Ω_{A} is the least positive real number in L_{ω} = −c_{A} · 2πiL_{f}, when the period lattice is not rectangular, and twice the least positive real number when it is.

17.2.7 The Special value L(A, 1)

Proposition 17.2.8. We have L(f, 1) ∈ R.

Proof. With the right setup, this would follow immediately from the fact that z ↦ −\overline{z} fixes the homology class \{0, ∞\}. However, we don’t have such a setup, so we give a direct proof.

Just as in the proof of the functional equation for Λ(f, s), use that f is an eigenvector for the Atkin-Lehner operator W_{N} and (17.2.4) to write L(f, 1) as the sum of two integrals from i/\sqrt{N} to i∞. Then use the calculation

\overline{2πi ∫_{i/\sqrt{N}}^{i∞} ∑_{n=1}^{∞} a_{n}e^{2πinz}dz} = −2πi ∑_{n=1}^{∞} a_{n}\overline{∫_{i/\sqrt{N}}^{i∞} e^{2πinz}dz}

= −2πi ∑_{n=1}^{∞} a_{n} \overline{\frac{1}{2πin}e^{−2πn/\sqrt{N}}}

= 2πi ∑_{n=1}^{∞} a_{n} \frac{1}{2πin}e^{2πn/\sqrt{N}}

to see that \overline{L(f, 1)} = L(f, 1).

Remark 17.2.9. The BSD conjecture implies that L(f, 1) ≥ 0, but this is unknown (it follows from GRH for L(f, s)).

17.2.8 Rationality of L(A, 1)/Ω_{A}

Proposition 17.2.10. Suppose A = A_{f} is an elliptic curve. Then L(A, 1)/Ω_{A} ∈ Q. More precisely, if n is the smallest multiple of {0, ∞} that lies in H_{1}(X_{0}(N), Z) and c_{A} is the Manin constant of A, then 2n · c_{A} · L(A, 1)/Ω_{A} ∈ Z.

Proof. By the Manin-Drinfeld theorem n{0, ∞} ∈ H_{1}(X_{0}(N), Z), so

n · L(f, 1) = −n · 2πi · ∫_{0}^{i∞} f(z)dz ∈ −2πi · L_{f} = \frac{1}{c_{A}} L_{ω}.

Combining this with Proposition 17.2.8, we see that

n · c_{A} · L(f, 1) ∈ L_{ω}^{+},

where L_{ω}^{+} is the submodule fixed by complex conjugation (i.e., L_{ω}^{+} = L∩R). When the period lattice is not rectangular, Ω_{A} generates L_{ω}^{+}, and when it is rectangular, \frac{1}{2}Ω_{A} generates. Thus n · c_{A} · L(f, 1) is an integer multiple of \frac{1}{2}Ω_{A}, which proves the proposition.

Proposition 17.2.10 can be more precise and generalized to abelian varieties A = A_{f}^{∨} attached to newforms. One can also replace n by the order of the image of (0) − (∞) in A(Q).

Theorem 17.2.11 (Agashe, Stein). Suppose f ∈ S_{2}(Γ_{1}(N)) is a newform and let A = A_{f}^{∨} be the abelian variety attached to f. Then we have the following equality of rational numbers:

\frac{|L(A, 1)|}{Ω_{A}} = \frac{1}{c_{∞} · c_{A}} · [Φ(H_{1}(X_{1}(N), Z))^{+} : Φ(T\{0, ∞\})].

Note that L(A, 1) ∈ R, so |L(A, 1)| = ±L(A, 1), and one expects, of course, that L(A, 1) ≥ 0.

For V and W lattices in an R-vector space M, the lattice index [V : W] is by definition the absolute value of the determinant of a change of basis taking a basis for V to a basis for W, or 0 if W has rank smaller than the dimension of M.

Proof. Let Ω̃_{A} be the measure of A(R) with respect to a basis for S_{2}(Γ_{1}(N), Z)[I_{f}], where I_{f} is the annihilator in T of f. Note that Ω̃_{A} · c_{A} = Ω_{A}, where c_{A} is the Manin index. Unwinding the definitions, we find that

Ω̃_{A} = c_{∞} · [Hom(S_{2}(Γ_{1}(N), Z)[I_{f}], Z) : Φ(H_{1}(X_{0}(N), Z))^{+}].

For any ring R the pairing^{4} 4

T_{R} × S_{2}(Γ_{1}(N), R) → R

given by ⟨T_{n}, f⟩ = a_{1}(T_{n}f) is perfect, so (T/I_{f}) ⊗ R ≅ Hom(S_{2}(Γ_{1}(N), R)[I_{f}], R). Using this pairing, we may view Φ as a map

Φ : H_{1}(X_{1}(N), Q) → (T/I_{f}) ⊗ C,

so that

Ω̃_{A} = c_{∞} · [T/I_{f} : Φ(H_{1}(X_{0}(N), Z))^{+}].

Note that (T/I_{f}) ⊗ C is isomorphic as a ring to a product of copies of C, with one copy corresponding to each Galois conjugate f_{i} of f. Let π_{i} ∈ (T/I_{f}) ⊗ C be the projector onto the subspace of (T/I_{f}) ⊗ C corresponding to f_{i}. Then

Φ({0, ∞}) · π_{i} = L(f_{i}, 1) · π_{i}.

Since the π_{i} form a basis for the complex vector space (T/I_{f}) ⊗ C, if we view Φ({0, ∞}) as the operator “left-multiplication by Φ({0, ∞})”, then

det(Φ({0, ∞})) = ∏_{i} L(f_{i}, 1) = L(A, 1),

Letting H = H_{1}(X_{0}(N), Z), we have

[Φ(H)^{+} : Φ(T{0, ∞})] = [Φ(H)^{+} : (T/I_{f}) · Φ({0, ∞})]

= [Φ(H)^{+} : T/I_{f}] · [T/I_{f} : T/I_{f} · Φ({0, ∞})]

= \frac{c_{∞}}{Ω̃_{A}} · | det(Φ(\{0, ∞\}))|

= \frac{c_{∞}c_{A}}{Ω_{A}} · |L(A, 1)|,

4reference!

which proves the theorem.

Remark 17.2.12. Theorem 17.2.11 is false, in general, when A is a quotient of J_{1}(N) not attached to a single Gal(\overline{Q}/Q)-orbit of newforms. It could be modified to handle this more general case, but the generalization seems not to has been written down.

17.3 General refined conjecture

Conjecture 17.3.1 (Birch and Swinnerton-Dyer). Let r = ord_{s=1} L(A, s). Then r is the rank of A(Q), the group � (A) is finite, and

\frac{L^{(r)}(A, 1)}{r!} = \frac{#� (A) · Ω_{A} · Reg_{A} · ∏_{p|N} c_{p}}{#A(Q)_{tor} · #A^{∨}(Q)_{tor}} .

17.4 The Conjecture for non-modular abelian varieties

Conjecture 17.3.1 can be extended to general abelian varieties over global fields. Here we discuss only the case of a general abelian variety A over Q. We follow the discussion in [Lan91, 95-94] (Lang, Number Theory III)^{5} , which describes Gross’s 5 formulation of the conjecture for abelian varieties over number fields, and to which we refer the reader for more details.

For each prime number ℓ, the ℓ-adic Tate module associated to A is

Ta_{ℓ}(A) = \underleftarrow{lim}_{n} A(\overline{Q})[ℓ^{n}].

Since A(\overline{Q})[ℓ^{n}] ≅ (Z/ℓ^{n}Z)^{2 dim(A)}, we see that Ta_{ℓ}(A) is free of rank 2 dim(A) as a Z_{ℓ}-module. Also, since the group structure on A is defined over Q, Ta_{ℓ}(A) comes equipped with an action of Gal(\overline{Q}/Q):

ρ_{A,ℓ} : Gal(\overline{Q}/Q) → Aut(Ta_{ℓ}(A)) ≈ GL_{2d}(Z_{ℓ}).

Suppose p is a prime and let ℓ ≠ p be another prime. Fix any embedding \overline{Q} ↪ \overline{Q}_{p}, and notice that restriction defines a homorphism r : Gal(\overline{Q}_{p}/Q_{p}) → Gal(\overline{Q}/Q). Let G_{p} ⊂ Gal(\overline{Q}/Q) be the image of r. The inertia group I_{p} ⊂ G_{p} is the kernel of the natural surjective reduction map, and we have an exact sequence

0 → I_{p} → Gal(\overline{Q}_{p}/Q_{p}) → Gal(\overline{F}_{p}/F_{p}) → 0.

The Galois group Gal(\overline{F}_{p}/F_{p}) is isomorphic to Ẑ with canonical generator x ↦ x^{p}. Lifting this generator, we obtain an element Frob_{p} ∈ Gal(\overline{Q}_{p}/Q_{p}), which is welldefined up to an element of I_{p}. Viewed as an element of G_{p} ⊂ Gal(\overline{Q}/Q), the element Frob_{p} is well-defined up I_{p} and our choice of embedding \overline{Q} ↪ \overline{Q}_{p}. One can show that this implies that Frob_{p} ∈ Gal(\overline{Q}/Q) is well-defined up to I_{p} and conjugation by an element of Gal(\overline{Q}/Q).

5Remove paren.

For a G_{p}-module M, let

M^{I_{p}} = {x ∈ M : σ(x) = x all σ ∈ I_{p}}.

Because I_{p} acts trivially on M^{I_{p}}, the action of the element Frob_{p} ∈ Gal(\overline{Q}/Q) on M^{I_{p}} is well-defined up to conjugation (I_{p} acts trivially, so the “up to I_{p}” obstruction vanishes). Thus the characteristic polynomial of Frob_{p} on M^{I_{p}} is welldefined, which is why L_{p}(A, s) is well-defined. The local L-factor of L(A, s) at p is

L_{p}(A, s) = \frac{1}{det (I − p^{−s} Frob_{p}^{−1} | Hom_{Z_{ℓ}}(Ta_{ℓ}(A), Z_{ℓ})^{I_{p}})}.

Definition 17.4.1. L(A, s) = ∏_{all p} L_{p}(A, s)

For all but finitely many primes Ta_{ℓ}(A)^{I_{p}} = Ta_{ℓ}(A). For example, if A = A_{f} is attached to a newform f = ∑ a_{n}q^{n} of level N and p ∤ ℓ·N, then Ta_{ℓ}(A)^{I_{p}} = Ta_{ℓ}(A). In this case, the Eichler-Shimura relation implies that L_{p}(A, s) equals ∏ L_{p}(f_{i}, s), where the f_{i} = ∑ a_{n,i}q^{n} are the Galois conjugates of f and L_{p}(f_{i}, s) = (1 − a_{p,i} · p^{−s} + p^{1−2s})^{−1}. The point is that Eichler-Shimura can be used to show that the characteristic polynomial of Frob_{p} is ∏_{i=1}^{dim(A)}(X^{2} − a_{p,i}X + p^{1−2s}).

Theorem 17.4.2. L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s).

17.5 Visibility of Shafarevich-Tate groups

Let K be a number field. Suppose

0 → A → B → C → 0

is an exact sequence of abelian varieties over K. (Thus each of A, B, and C is a complete group variety over K, whose group is automatically abelian.) Then there is a corresponding long exact sequence of cohomology for the group Gal(\overline{Q}/K):

0 → A(K) → B(K) → C(K) ⟶^{δ} H^{1}(K, A) → H^{1}(K, B) → H^{1}(K, C) → ⋯

The study of the Mordell-Weil group C(K) = H^{0}(K, C) is popular in arithmetic geometry. For example, the Birch and Swinnerton-Dyer conjecture (BSD conjecture), which is one of the million dollar Clay Math Problems, asserts that the dimension of C(K) ⊗ Q equals the ordering vanishing of L(C, s) at s = 1.

The group H^{1}(K, A) is also of interest in connection with the BSD conjecture, because it contains the Shafarevich-Tate group

� (A) = � (A/K) = Ker (H^{1}(K, A) → ⨁_{v} H^{1}(K_{v}, A)) ⊂ H^{1}(K, A),

where the sum is over all places v of K (e.g., when K = Q, the fields K_{v} are Q_{p} for all prime numbers p and Q_{∞} = R).

The group A(K) is fundamentally different than H^{1}(K, C). The Mordell-Weil group A(K) is finitely generated, whereas the first Galois cohomology H^{1}(K, C) is far from being finitely generated—in fact, every element has finite order and there are infinitely many elements of any given order.

This talk is about “dimension shifting”, i.e., relating information about H^{0}(K, C) to information about H^{1}(K, A).

17.5.1 Definitions

Elements of H^{0}(K, C) are simply points, i.e., elements of C(K), so they are relatively easy to “visualize”. In contrast, elements of H^{1}(K, A) are Galois cohomology classes, i.e., equivalence classes of set-theoretic (continuous) maps f : Gal(\overline{Q}/K) → A(\overline{Q}) such that f(στ) = f(σ) + σf(τ). Two maps are equivalent if their difference is a map of the form σ ↦ σ(P) − P for some fixed P ∈ A(\overline{Q}). From this point of view H^{1} is more mysterious than H^{0}.

There is an alternative way to view elements of H^{1}(K, A). The WC group of A is the group of isomorphism classes of principal homogeneous spaces for A, where a principal homogeneous space is a variety X and a map A × X → X that satisfies the same axioms as those for a simply transitive group action. Thus X is a twist as variety of A, but X(K) = ∅, unless X ≈ A. Also, the nontrivial elements of � (A) correspond to the classes in WC that have a K_{v}-rational point for all places v, but no K-rational point.

Mazur introduced the following definition in order to help unify diverse constructions of principal homogeneous spaces:

Definition 17.5.1 (Visible). The visible subgroup of H^{1}(K, A) in B is

Vis_{B} H^{1}(K, A) = Ker(H^{1}(K, A) → H^{1}(K, B))
= Coker(B(K) → C(K)).

Remark 17.5.2. Note that Vis_{B} H^{1}(K, A) does depend on the embedding of A into B. For example, suppose B = B_{1} × A. Then there could be nonzero visible elements if A is embedding into the first factor, but there will be no nonzero visible elements if A is embedded into the second factor. Here we are using that H^{1}(K, B_{1} × A) = H^{1}(K, B_{1}) ⊕ H^{1}(K, A).

The connection with the WC group of A is as follows. Suppose

0 → A ⟶^{f} B ⟶^{g} C → 0

is an exact sequence of abelian varieties and that c ∈ H^{1}(K, A) is visible in B. Thus there exists x ∈ C(K) such that δ(x) = c, where δ : C(K) → H^{1}(K, A) is the connecting homomorphism. Then X = π^{−1}(x) ⊂ B is a translate of A in B, so the group law on B gives X the structure of principal homogeneous space for A, and one can show that the class of X in the WC group of A corresponds to c.

Lemma 17.5.3. The group Vis_{B} H^{1}(K, A) is finite.

Proof. Since Vis_{B} H^{1}(K, A) is a homomorphic image of the finitely generated group C(K), it is also finitely generated. On the other hand, it is a subgroup of H^{1}(K, A), so it is a torsion group. The lemma follows since a finitely generated torsion abelian group is finite.

17.5.2 Every element of H^{1}(K, A) is visible somewhere

Proposition 17.5.4. Let c ∈ H^{1}(K, A). Then there exists an abelian variety B = B_{c} and an embedding A ↪ B such that c is visible in B.

Proof. By definition of Galois cohomology, there is a finite extension L of K such that res_{L}(c) = 0. Thus c maps to 0 in H^{1}(L, A_{L}). By a slight generalization of the Shapiro Lemma from group cohomology (which can be proved by dimension shifting; see, e.g.,^{6} Atiyah-Wall in Cassels-Frohlich), there is a canonical isomorphism 6

H^{1}(L, A_{L}) ≅ H^{1}(K, Res_{L/K}(A_{L})) = H^{1}(K, B),

where B = Res_{L/K}(A_{L}) is the Weil restriction of scalars of A_{L} back down to K. The restriction of scalars B is an abelian variety of dimension [L : K] · dim A that is characterized by the existence of functorial isomorphisms

Mor_{K}(S, B) ≅ Mor_{L}(S_{L}, A_{L}),

for any K-scheme S, i.e., B(S) = A_{L}(S_{L}). In particular, setting S = A we find that the identity map A_{L} → A_{L} corresponds to an injection A ↪ B. Moreover, c ↦ res_{L}(c) = 0 ∈ H^{1}(K, B).

Remark 17.5.5. The abelian variety B in Proposition 17.5.4 is a twist of a power of A.

17.5.3 Visibility in the context of modularity

Usually we focus on visibility of elements in � (A). There are a number of other results about visibility in various special cases, and large tables of examples in the context of elliptic curves and modular abelian varieties. There are also interesting modularity questions and conjectures in this context.

Motivated by the desire to understand the Birch and Swinnerton-Dyer conjecture more explicitly, I developed^{7} (with significant input from Agashe, Cremona, Mazur, 7 and Merel) computational techniques for unconditionally constructing Shafarevich-Tate groups of modular abelian varieties A ⊂ J_{0}(N) (or J_{1}(N)). For example, if A ⊂ J_{0}(389) is the 20-dimensional simple factor, then

Z/5Z × Z/5Z ⊂ � (A),

as predicted by the Birch and Swinnerton-Dyer conjecture. See [CM00] for examples when dim A = 1. We will spend the rest of this section discussing the examples of [AS05, AS02] in more detail.

Tables 17.5.1–17.5.4 illustrate the main computational results of [AS05]. These tables were made by gathering data about certain arithmetic invariants of the 19608 abelian varieties A_{f} of level ≤ 2333. Of these, exactly 10360 have satisfy L(A_{f}, 1) ≠ 0, and for these with L(A_{f}, 1) ≠ 0, we compute a divisor and multiple of the conjectural order of � (A_{f}). We find that there are at least 168 such that the Birch and Swinnerton-Dyer Conjecture implies that � (A_{f}) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of � (A_{f}) really divides #� (A_{f}) by constructing nontrivial elements of � (A_{f}) using visibility.

6Fix

7change.

The meaning of the tables is as follows. The first column lists a level N and an isogeny class, which uniquely specifies an abelian variety A = A_{f} ⊂ J_{0}(N). The nth isogeny class is given by the nth letter of the alphabet. We will not discuss the ordering further, except to note that usually, the dimension of A, which is given in the second column, is enough to determine A. When L(A, 1) ≠ 0, Conjecture 17.2.1 predicts that

#� (A) =^{?} \frac{L(A, 1)}{Ω_{A}} · \frac{#A(Q)_{tor} · #A^{∨}(Q)_{tor}}{∏_{p|N} c_{p}} .

We view the quotient L(A, 1)/Ω_{A}, which is a rational number, as a single quantity. We can compute multiples and divisors of every quantity appearing in the right hand side of this equation, and this yields columns three and four, which are a divisor S_{ℓ} and a multiple S_{u} of the conjectural order of � (A) (when S_{u} = S_{ℓ}, we put an equals sign in the S_{u} column). Column five, which is labeled odd deg(φ_{A}), contains the odd part of the degree of the polarization

φ_{A} : (A ↪ J_{0}(N) ≅ J_{0}(N)^{∨} → A^{∨}).  (17.5.1)

The second set of columns, columns six and seven, contain an abelian variety B = B_{g} ⊂ J_{0}(N) such that #(A ∩ B) is divisible by an odd prime divisor of S_{ℓ} and L(B, 1) = 0. When dim(B) = 1, we have verified that B is an elliptic curve of rank 2. The eighth column A ∩ B contains the group structure of A ∩ B, where e.g., [2^{2}302^{2}] is shorthand notation for (Z/2Z)^{2} ⊕ (Z/302Z)^{2}. The final column, labeled Vis, contains a divisor of the order of Vis_{A+B}(� (A)).

The following proposition explains the significance of the odd deg(φ_{A}) column.

Proposition 17.5.6. If p ∤ deg(φ_{A}), then p ∤ Vis_{J_{0}(N)}(H^{1}(Q, A)).

Proof. There exists a complementary morphism φ̂_{A}, such that φ_{A}∘φ̂_{A} = φ̂_{A}∘φ_{A} = [n], where n is the degree of φ_{A}. If c ∈ H^{1}(Q, A) maps to 0 in H^{1}(Q, J_{0}(N)), then it also maps to 0 under the following composition

H^{1}(Q, A) → H^{1}(Q, J_{0}(N)) → H^{1}(Q, A^{∨}) ⟶^{φ̂_{A}} H^{1}(Q, A).

Since this composition is [n], it follows that c ∈ H^{1}(Q, A)[n], which proves the proposition.

Remark 17.5.7. Since the degree of φ_{A} does not change if we extend scalars to a number field K, the subgroup of H^{1}(K, A) visible in J_{0}(N)_{K}, still has order divisible only by primes that divide deg(φ_{A}).

The following theorem explains the significance of the B column, and how it was used to deduce the Vis column.

Theorem 17.5.8. Suppose A and B are abelian subvarieties of an abelian variety C over Q and that A(\overline{Q}) ∩ B(\overline{Q}) is finite. Assume also that A(Q) is finite. Let N be an integer divisible by the residue characteristics of primes of bad reduction for C (e.g., N could be the conductor of C). Suppose p is a prime such that

p ∤ 2 · N · #((A + B)/B)(Q)_{tor} · #B(Q)_{tor} · ∏_{ℓ} c_{A,ℓ} · c_{B,ℓ},

where c_{A,ℓ} = #Φ_{A,ℓ}(F_{ℓ}) is the Tamagawa number of A at ℓ (and similarly for B). Suppose furthermore that B(\overline{Q})[p] ⊂ A(\overline{Q}) as subgroups of C(\overline{Q}). Then there is a natural injection

B(Q)/pB(Q) ↪ Vis_{C}(� (A)).

A complete proof of a generalization of this theorem can be found in [AS02].

Sketch of Proof. Without loss of generality, we may assume C = A + B. Our hypotheses yield a diagram

\xymatrix{0 \ar[r] & B[p] \ar[r] \ar[d] & B \ar[r]^{p} \ar[d] & B \ar[r] \ar@{.>}[d] & 0 \\ 0 \ar[r] & A \ar[r] & C \ar[r] & B′ \ar[r] & 0,}

where B′ = C/A. Taking Gal(\overline{Q}/Q)-cohomology, we obtain the following diagram:

\xymatrix{0 \ar[r] & B(Q) \ar[r]^{p} \ar[d] & B(Q) \ar[r] \ar[d] & B(Q)/pB(Q) \ar[r] \ar@{.>}[d] & 0 \\ 0 \ar[r] & C(Q)/A(Q) \ar[r] & B′(Q) \ar[r] & Vis_{C}(H^{1}(Q, A)) \ar[r] & 0.}

The snake lemma and our hypothesis that p ∤ #(C/B)(Q)_{tor} imply that the rightmost vertical map is an injection

i : B(Q)/pB(Q) ↪ Vis_{C}(H^{1}(Q, A)),  (17.5.2)

since C(A)/(A(Q) + B(Q)) is a sub-quotient of (C′/B)(Q).

We show that the image of (17.5.2) lies in � (A) using a local analysis at each prime, which we now sketch. At the archimedian prime, no work is needed since p ≠ 2. At non-archimedian primes ℓ, one uses facts about Néron models (when ℓ = p) and our hypothesis that p does not divide the Tamagawa numbers of B (when ℓ ≠ p) to show that if x ∈ B(Q)/pB(Q), then the corresponding cohomology class res_{ℓ}(i(x)) ∈ H^{1}(Q_{ℓ}, A) splits over the maximal unramified extension. However,

H^{1}(Q_{ℓ}^{ur}/Q_{ℓ}, A) ≅ H^{1}(\overline{F}_{ℓ}/F_{ℓ}, Φ_{A,ℓ}(\overline{F}_{ℓ})),

and the right hand cohomology group has order c_{A,ℓ}, which is coprime to p. Thus res_{ℓ}(i(x)) = 0, which completes the sketch of the proof.

17.5.4 Future directions

The data in Tables 17.5.1-17.5.4 could be investigated further.

It should be possible to replace the hypothesis that B[p] ⊂ A, with the weaker hypothesis that B[𝔪] ⊂ A, where 𝔪 is a maximal ideal of the Hecke algebra T. For example, this improvement would help one to show that 5^{2} divides the order of the Shafarevich-Tate group of 1041E. Note that for this example, we only know that L(B, 1) = 0, not that B(Q) has positive rank (as predicted by Conjecture 17.1.5), which is another obstruction.

One can consider visibility at a higher level. For example, there are elements of order 3 in the Shafarevich-Tate group of 551H that are not visible in J_{0}(551), but these elements are visible in J_{0}(2 · 551), according to the computations in

[Ste04] (Studying the Birch and Swinnerton-Dyer Conjecture for Modular Abelian Varieties Using MAGMA).^{8} 8

Conjecture 17.5.9 (Stein). Suppose c ∈ � (A_{f}), where A_{f} ⊂ J_{0}(N). Then there exists M such that c is visible in J_{0}(NM). In other words, every element of � (A_{f}) is “modular”.

17.6 Kolyvagin’s Euler system of Heegner points

In this section we will briefly sketch some of the key ideas behind Kolyvagin’s proof of Theorem 17.1.9. We will follow [Rub89] very closely. Two other excellent references are [Gro91] and [McC91]. Kolyvagin’s original papers^{9} on this theorem 9 are not so easy to read because they are all translations from Russian, but none of the three papers cited above give a complete proof of his theorem.

We only sketch a proof of the following special case of Kolyvagin’s theorem.

Theorem 17.6.1. Suppose E is an elliptic curve over Q such that L(E, 1) ≠ 0. Then E(Q) is finite and � (E)[p] = 0 for almost all primes p.

The strategy of the proof is as follows. Applying Galois cohomology to the multiplication by p sequence

0 → E[p] → E ⟶^{p} E → 0,

we obtain a short exact sequence

0 → E(Q)/pE(Q) → H^{1}(Q, E[p]) → H^{1}(Q, E)[p] → 0.

The inverse image of � (E)[p] in H^{1}(Q, E[p]) is called the Selmer group of E at p, and we will denote it by Sel^{(p)}(E). We have an exact sequence

0 → E(Q)/pE(Q) → Sel^{(p)}(E) → � (E)[p] → 0.

Because E(Q) is finitely generated, to prove Theorem 17.6.1 it suffices to prove that Sel^{(p)}(E) = 0 for all but finitely many primes p. We do this by using complex multiplication elliptic curves to construct Galois cohomology classes with precise local behavior.

It is usually easier to say something about the Galois cohomology of a module over a local field than over Q, so we introduce some more notation to help formalize this. For any prime p and place v we have a commutative diagram

\xymatrix{0 \ar[r] & E(Q)/pE(Q) \ar[r] \ar[d] & H^{1}(Q, E[p]) \ar[r] \ar[d]^{_{res_{v}}} & H^{1}(Q, E)[p] \ar[r] \ar[d]^{_{res_{v}}} & 0 \\ 0 \ar[r] & E(Q_{v})/pE(Q_{v}) \ar[r] & H^{1}(Q_{v}, E[p]) \ar[r] & H^{1}(Q_{v}, E)[p] \ar[r] & 0}

8remove

9Add references.

168 17. The Birch and Swinnerton-Dyer Conjecture

TABLE 17.5.1. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim S_{l}  S_{u} odd deg(φ_{A})  B dim A ∩ B Vis
389E∗ 20 5^{2}  =  5  389A 1 [20^{2}] 5^{2}
433D∗ 16 7^{2}  =  7·111  433A 1 [14^{2}] 7^{2}
446F∗ 8 11^{2}  = 11·359353  446B 1 [11^{2}] 11^{2}
551H 18 3^{2}  =  169  NONE
563E∗ 31 13^{2}  =  13  563A 1 [26^{2}] 13^{2}
571D∗ 2  3^{2}  =  3^{2}·127  571B 1  [3^{2}]  3^{2}
655D∗ 13 3^{4}  = 3^{2}·9799079  655A 1 [36^{2}] 3^{4}
681B 1  3^{2}  =  3·125  681C 1  [3^{2}]  −
707G∗ 15 13^{2}  = 13·800077  707A 1 [13^{2}] 13^{2}
709C∗ 30 11^{2}  =  11  709A 1 [22^{2}] 11^{2}
718F∗ 7  7^{2}  = 7·5371523  718B 1  [7^{2}]  7^{2}
767F 23 3^{2}  =  1  NONE
794G 12 11^{2}  = 11·34986189 794A 1 [11^{2}]  −
817E 15 7^{2}  =  7·79  817A 1  [7^{2}]  −
959D 24 3^{2}  =  583673  NONE
997H∗ 42 3^{4}  =  3^{2}  997B 1 [12^{2}] 3^{2}
997C 1 [24^{2}] 3^{2}
1001F 3  3^{2}  =  3^{2}·1269  1001C 1  [3^{2}]  −
91A 1  [3^{2}]  −
1001L 7  7^{2}  = 7·2029789  1001C 1  [7^{2}]  −
1041E 4  5^{2}  =  5^{2}·13589  1041B 2  [5^{2}]  −
1041J 13 5^{4}  = 5^{3}·21120929983 1041B 2  [5^{4}]  −
1058D 1  5^{2}  =  5·483  1058C 1  [5^{2}]  −
1061D 46 151^{2} = 151·10919  1061B 2 [2^{2}302^{2}] −
1070M 7 3·5^{2} 3^{2}·5^{2} 3·5·1720261 1070A 1 [15^{2}]  −
1077J 15 3^{4}  = 3^{2}·1227767047943 1077A 1  [9^{2}]  −
1091C 62 7^{2}  =  1  NONE
1094F∗ 13 11^{2}  = 11^{2}·172446773 1094A 1 [11^{2}] 11^{2}
1102K 4 3^{2} = 3^{2}·31009 1102A 1 [3^{2}] −
1126F∗ 11 11^{2}  = 11·13990352759 1126A 1 [11^{2}] 11^{2}
1137C 14 3^{4}  = 3^{2}·64082807 1137A 1  [9^{2}]  −
1141I 22 7^{2}  =  7·528921  1141A 1 [14^{2}]  −
1147H 23 5^{2}  =  5·729  1147A 1 [10^{2}]  −
1171D∗ 53 11^{2}  =  11·81  1171A 1 [44^{2}] 11^{2}
1246B 1  5^{2}  =  5·81  1246C 1  [5^{2}]  −
1247D 32 3^{2}  =  3^{2}·2399  43A 1 [36^{2}]  −
1283C 62 5^{2}  =  5·2419  NONE
1337E 33 3^{2}  =  71  NONE
1339G 30 3^{2}  =  5776049  NONE
1355E 28  3  3^{2} 3^{2}·2224523985405 NONE
1363F 25 31^{2}  =  31·34889  1363B 2 [2^{2}62^{2}] −
1429B 64 5^{2}  =  1  NONE
1443G 5  7^{2}  =  7^{2}·18525  1443C 1 [7^{1}14^{1}] −
1446N 7  3^{2}  = 3·17459029  1446A 1 [12^{2}]  −

17.6 Kolyvagin’s Euler system of Heegner points 169

TABLE 17.5.2. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim S_{l} S_{u} odd deg(φ_{A})  B dim A ∩ B Vis
1466H∗ 23  13^{2} = 13·25631993723 1466B 1  [26^{2}]  13^{2}
1477C∗ 24  13^{2} = 13·57037637  1477A 1  [13^{2}]  13^{2}
1481C 71  13^{2} =  70825  NONE
1483D∗ 67 3^{2}·5^{2} =  3·5  1483A 1  [60^{2}] 3^{2}·5^{2}
1513F 31  3  3^{4}  3·759709  NONE
1529D 36  5^{2}  =  535641763  NONE
1531D 73  3  3^{2}  3  1531A 1  [48^{2}]  −
1534J 6  3  3^{2} 3^{2}·635931  1534B 1  [6^{2}]  −
1551G 13  3^{2}  = 3·110659885  141A 1  [15^{2}]  −
1559B 90  11^{2} =  1  NONE
1567D 69 7^{2}·41^{2} =  7·41  1567B 3 [4^{4}1148^{2}] −
1570J∗ 6  11^{2} = 11·228651397  1570B 1  [11^{2}]  11^{2}
1577E 36  3  3^{2}  3^{2}·15  83A 1  [6^{2}]  −
1589D 35  3^{2}  = 6005292627343  NONE
1591F∗ 35  31^{2} =  31·2401  1591A 1  [31^{2}]  31^{2}
1594J 17  3^{2}  = 3·259338050025131 1594A 1  [12^{2}]  −
1613D∗ 75  5^{2}  =  5·19  1613A 1  [20^{2}]  5^{2}
1615J 13  3^{4}  = 3^{2}·13317421  1615A 1 [9^{1}18^{1}]  −
1621C∗ 70  17^{2} =  17  1621A 1  [34^{2}]  17^{2}
1627C∗ 73  3^{4}  =  3^{2}  1627A 1  [36^{2}]  3^{4}
1631C 37  5^{2}  = 6354841131  NONE
1633D 27 3^{6}·7^{2} = 3^{5}·7·31375  1633A 3 [6^{4}42^{2}]  −
1634K 12  3^{2}  = 3·3311565989  817A 1  [3^{2}]  −
1639G∗ 34  17^{2} =  17·82355  1639B 1  [34^{2}]  17^{2}
1641J∗ 24  23^{2} = 23·1491344147471 1641B 1  [23^{2}]  23^{2}
1642D∗ 14  7^{2}  = 7·123398360851  1642A 1  [7^{2}]  7^{2}
1662K 7  11^{2} = 11·16610917393 1662A 1  [11^{2}]  −
1664K 1  5^{2}  =  5·7  1664N 1  [5^{2}]  −
1679C 45  11^{2} =  6489  NONE
1689E 28  3^{2}  = 3·172707180029157365 563A 1  [3^{2}]  −
1693C 72 1301^{2} =  1301  1693A 3 [2^{4}2602^{2}] −
1717H∗ 34  13^{2} =  13·345  1717B 1  [26^{2}]  13^{2}
1727E 39  3^{2}  =  118242943  NONE
1739F 43 659^{2} = 659·151291281  1739C 2 [2^{2}1318^{2}] −
1745K 33  5^{2}  = 5·1971380677489 1745D 1  [20^{2}]  −
1751C 45  5^{2}  =  5·707  103A 2 [505^{2}]  −
1781D 44  3^{2}  =  61541  NONE
1793G∗ 36  23^{2} = 23·8846589  1793B 1  [23^{2}]  23^{2}
1799D 44  5^{2}  =  201449  NONE
1811D 98  31^{2} =  1  NONE
1829E 44  13^{2} =  3595  NONE
1843F 40  3^{2}  =  8389  NONE
1847B 98  3^{6}  =  1  NONE
1871C 98  19^{2} =  14699  NONE

170 17. The Birch and Swinnerton-Dyer Conjecture TABLE 17.5.3. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim  S_{l}  S_{u}  odd deg(φ_{A})  B dim  A ∩ B  Vis
1877B 86  7^{2}  =  1  NONE
1887J 12  5^{2}  =  5·10825598693  1887A 1  [20^{2}]  −
1891H 40  7^{4}  =  7^{2}·44082137  1891C 2  [4^{2}196^{2}]  −
1907D∗ 90  7^{2}  =  7·165  1907A 1  [56^{2}]  7^{2}
1909D∗ 38  3^{4}  =  3^{2}·9317  1909A 1  [18^{2}]  3^{4}
1913B∗ 1  3^{2}  =  3·103  1913A 1  [3^{2}]  3^{2}
1913E 84 5^{4}·61^{2}  =  5^{2}·61·103  1913A 1  [10^{2}]  −
1913C 2  [2^{2}610^{2}]  −
1919D 52  23^{2}  =  675  NONE
1927E 45  3^{2}  3^{4}  52667  NONE
1933C 83  3^{2}·7 3^{2}·7^{2}  3·7  1933A 1  [42^{2}]  3^{2}
1943E 46  13^{2}  =  62931125  NONE
1945E∗ 34  3^{2}  =  3·571255479184807  389A 1  [3^{2}]  3^{2}
1957E∗ 37 7^{2}·11^{2}  =  7·11·3481  1957A 1  [22^{2}]  11^{2}
1957B 1  [14^{2}]  7^{2}
1979C 104  19^{2}  =  55  NONE
1991C 49  7^{2}  =  1634403663  NONE
1994D 26  3  3^{2}  3^{2}·46197281414642501  997B 1  [3^{2}]  −
1997C 93  17^{2}  =  1  NONE
2001L 11  3^{2}  =  3^{2}·44513447  NONE
2006E 1  3^{2}  =  3·805  2006D 1  [3^{2}]  −
2014L 12  3^{2}  =  3^{2}·126381129003  106A 1  [9^{2}]  −
2021E 50  5^{6}  =  5^{2}·729  2021A 1  [100^{2}]  5^{4}
2027C∗ 94  29^{2}  =  29  2027A 1  [58^{2}]  29^{2}
2029C 90 5^{2}·269^{2} =  5·269  2029A 2  [2^{2}2690^{2}]  −
2031H∗ 36  11^{2}  =  11·1014875952355  2031C 1  [44^{2}]  11^{2}
2035K 16  11^{2}  =  11·218702421  2035C 1  [11^{1}22^{1}]  −
2038F 25  5  5^{2}  5^{2}·92198576587  2038A 1  [20^{2}]  −
1019B 1  [5^{2}]  −
2039F 99 3^{4}·5^{2}  =  13741381043009  NONE
2041C 43  3^{4}  =  61889617  NONE
2045I 39  3^{4}  =  3^{3}·3123399893  2045C 1  [18^{2}]  −
409A 13 [9370199679^{2}] −
2049D 31  3^{2}  =  29174705448000469937  NONE
2051D 45  7^{2}  =  7·674652424406369  2051A 1  [56^{2}]  −
2059E 45  5·7^{2} 5^{2}·7^{2}  5^{2}·7·167359757  2059A 1  [70^{2}]  −
2063C 106  13^{2}  =  8479  NONE
2071F 48  13^{2}  =  36348745  NONE
2099B 106  3^{2}  =  1  NONE
2101F 46  5^{2}  =  5·11521429  191A 2  [155^{2}]  −
2103E 37 3^{2}·11^{2}  = 3^{2}·11·874412923071571792611 2103B 1  [33^{2}]  11^{2}
2111B 112 211^{2}  =  1  NONE
2113B 91  7^{2}  =  1  NONE
2117E∗ 45  19^{2}  =  19·1078389  2117A 1  [38^{2}]  19^{2}

17.6 Kolyvagin’s Euler system of Heegner points 171

TABLE 17.5.4. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim  S_{l}  S_{u}  odd deg(φ_{A})  B dim  A ∩ B  Vis
2119C 48  7^{2}  =  89746579  NONE
2127D 34  3^{2}  = 3·18740561792121901 709A 1  [3^{2}]  −
2129B 102  3^{2}  =  1  NONE
2130Y 4  7^{2}  =  7·83927  2130B 1  [14^{2}]  −
2131B 101 17^{2}  =  1  NONE
2134J 11  3^{2}  =  1710248025389  NONE
2146J 10  7^{2}  =  7·1672443  2146A 1  [7^{2}]  −
2159E 57  13^{2}  =  31154538351  NONE
2159D 56  3^{4}  =  233801  NONE
2161C 98  23^{2}  =  1  NONE
2162H 14  3  3^{2}  3·6578391763  NONE
2171E 54  13^{2}  =  271  NONE
2173H 44 199^{2}  =  199·3581  2173D 2  [398^{2}]  −
2173F 43  19^{2} 3^{2}·19^{2} 3^{2}·19·229341  2173A 1  [38^{2}]  19^{2}
2174F 31  5^{2}  = 5·21555702093188316107 NONE
2181E 27  7^{2}  = 7·7217996450474835 2181A 1  [28^{2}]  −
2193K 17  3^{2}  =  3·15096035814223  129A 1  [21^{2}]  −
2199C 36  7^{2}  = 7^{2}·13033437060276603 NONE
2213C 101  3^{4}  =  19  NONE
2215F 46  13^{2}  =  13·1182141633  2215A 1  [52^{2}]  −
2224R 11  79^{2}  =  79  2224G 2  [79^{2}]  −
2227E 51  11^{2}  =  259  NONE
2231D 60  47^{2}  =  91109  NONE
2239B 110 11^{4}  =  1  NONE
2251E∗ 99  37^{2}  =  37  2251A 1  [74^{2}]  37^{2}
2253C∗ 27  13^{2}  = 13·14987929400988647 2253A 1  [26^{2}]  13^{2}
2255J 23  7^{2}  =  15666366543129  NONE
2257H 46 3^{6}·29^{2}  =  3^{3}·29·175  2257A 1  [9^{2}]  −
2257D 2  [2^{2}174^{2}]  −
2264J 22  73^{2}  =  73  2264B 2  [146^{2}]  −
2265U 14  7^{2}  = 7^{2}·73023816368925 2265B 1  [7^{2}]  −
2271I∗ 43  23^{2}  = 23·392918345997771783 2271C 1  [46^{2}]  23^{2}
2273C 105  7^{2}  =  7^{2}  NONE
2279D 61  13^{2}  =  96991  NONE
2279C 58  5^{2}  =  1777847  NONE
2285E 45 151^{2}  = 151·138908751161 2285A 2  [2^{2}302^{2}]  −
2287B 109 71^{2}  =  1  NONE
2291C 52  3^{2}  =  427943  NONE
2293C 96 479^{2}  =  479  2293A 2  [2^{2}958^{2}]  −
2294F 15  3^{2}  =  3·6289390462793  1147A 1  [3^{2}]  −
2311B 110  5^{2}  =  1  NONE
2315I 51  3^{2}  =  3·4475437589723  463A 16 [13426312769169^{2}] −
2333C 101 83341^{2}  =  83341  2333A 4  [2^{6}166682^{2}]  −

Observe that

Sel^{(p)}(E) = ⋂_{v} res_{v}^{−1}( image E(Q_{v})).

For s ∈ Sel^{(p)}(E), let s_{v} denote the inverse image of res_{v}(s) in E(Q_{v})/pE(Q_{v}).

Our first proposition asserts that if we can construct a cohomology class with certain properties, then that cohomology class forces Sel^{(p)}(E) to be locally trivial.

Proposition 17.6.2. Suppose ℓ is a prime such that E(Q_{ℓ})[p] ≅ Z/pZ and suppose there is a cohomology class c_{ℓ} ∈ H^{1}(Q, E)[p] such that res_{v}(c_{ℓ}) ≠ 0 if and only if v = ℓ. Then res_{ℓ}(Sel^{(p)}(E)) = 0 ⊂ H^{1}(Q_{ℓ}, E[p]).

Sketch of proof. Suppose s ∈ Sel^{(p)}(E). Using the local Tate pairing we obtain, for each v, a nondegenerate pairing

⟨ , ⟩_{v} : E(Q_{v})/pE(Q_{v}) × H^{1}(Q_{v}, E)[p] → Z/pZ.

Unwinding the definitions, and using the fact that the sum of the local invariants of an element of Br(Q) is trivial, we see that our hypothesis that all but one restriction of c_{ℓ} is trivial implies that ⟨s_{ℓ}, res_{ℓ}(c_{ℓ})⟩ = 0. Since res_{ℓ}(c_{ℓ}) ≠ 0, this implies that s_{ℓ} = 0.

Let τ ∈ Gal(\overline{Q}/Q) denote a fixed choice of complex conjugation, which is defined by fixing an embedding of \overline{Q} into C. For any module M on which τ acts, let

M^{+} = {x ∈ M : τ(x) = x}  and  M^{−} = {x ∈ M : τ(x) = −x}.

If E is defined by y^{2} = x^{3} + ax + b and K = Q(\sqrt{d}), then the twist of E by K is the elliptic curve dy^{2} = x^{3} + ax + b.

Theorem 17.6.3. There exists infinitely many quadratic imaginary fields K in which all primes dividing the conductor N of E split, and such that L′(E^{K}, 1) ≠ 0, where E^{K} is the twist of E over K.

Fix a K as in the theorem such that O_{K}^{∗} = {±1}, i.e., so that the discriminant D of K is not −3 or −4.

Lemma 17.6.4. Suppose ℓ ∤ pDN and Frob_{ℓ}(K(E[p])/Q) = [τ]. Then ℓ is inert in K, we have a_{ℓ} ≡ ℓ + 1 ≡ 0 (mod p), and the four groups E(Q_{ℓ})[p], E(F_{ℓ})[p], E(K_{ℓ})[p]^{−}, and E(F_{ℓ^{2}})[p]^{−} are all cyclic of order p.

Proof. The first assertion is true because τ|_{K} has order 2. The characteristic polynomial of Frob_{ℓ}(K(E[p])/Q) on E[p] is x^{2} − a_{ℓ}x + ℓ, and the characteristic polynomial of τ on E[p] is x^{2} − 1, which proves the second assertion. For the third, we have

(Z/pZ)^{2} ≅ E(\overline{Q})[p] ≅ E(K_{ℓ})[p] ≅ E(Q_{ℓ})[p] ⊕ E(K_{ℓ})[p]^{−},

and each summand on the right must be nonzero since τ ≠ 1 on E[p].

For the rest of the argument, we assume that p is odd and does not divide the order of the class group of K. We will now use the theory of complex multiplication elliptic curves and class field theory to construct cohomology classes c_{ℓ} which we will use in Proposition 17.6.2.

Since every prime dividing the conductor N of E splits in K, there is an ideal N of K such that O_{K}/N ≅ Z/NZ. Fix such an ideal for the rest of the argument. Let

N^{−1} = {x ∈ K : xN ⊂ O_{K}}

be the inverse of N in the group of fractional ideals of O_{K}. The pair

(C/O_{K}, N^{−1}/O_{K})

then defines a point on x ∈ X_{0}(N). By complex multiplication theory, the point x is defined over the Hilbert Class Field H of K, which is the maximal unramified abelian extension of K. Also, class field theory implies that Gal(H/K) is isomorphic to the ideal class group of K.

Let π : X_{0}(N) → E be a (minimal) modular parametrization. Then y_{K} = Tr_{H/K} π(x_{H}) ∈ E(K), and we call

y = y_{K} − τ(y_{K}) ∈ E(K)^{−}

the Heegner point associated to K. Our hypotheses on K imply, by the theorem of Gross and Zagier, that y has infinite order, a fact which will be crucial in our construction of cohomology classes c_{ℓ}. If we were to choose a K such that y were torsion (of order coprime to p), then our classes c_{ℓ} would be locally trivial, hence give candidate elements of � (E) (see [Gro91, §11, pg. 254] and [McC91] for more on this connection).

Suppose ℓ ∤ N is inert in K (i.e., does not split). Let

O_{ℓ} = Z + ℓO_{K},

which is the order of conductor ℓ in O_{K}. Notice that O_{K}/O_{ℓ} ≅ Z/ℓZ as groups. Since K ⊂ C, it make sense to view O_{ℓ} as a lattice in C. The pair

(C/O_{ℓ}, (N ∩ O_{ℓ})^{−1}/O_{ℓ})

then defines a CM point x_{ℓ} ∈ X_{0}(N). This point won’t be defined over the Hilbert class field H, though, but instead over an abelian extension K[ℓ] of K that is a cyclic extension of H of degree ℓ + 1 totally ramified over H at ℓ and unramified everywhere else. Let

y_{ℓ} = π(x_{ℓ}) ∈ E(K[ℓ]).

Proposition 17.6.5. We have Tr_{K[ℓ]/H}(y_{ℓ}) = a_{ℓ}y_{H}, where y_{H} = π(x_{H}). Also, for any prime λ of K[ℓ] lying above ℓ, we have

ỹ_{ℓ} = Frob_{ℓ}(ỹ_{H}),

as elements of E(F_{ℓ^{2}}).

We will not prove this proposition, except to note that it follows from the explicit descriptions of the action of Gal(\overline{Q}/Q) on CM points on X_{0}(N) and of the Hecke correspondences.

We are almost ready to construct the cohomology classes c_{ℓ}. Suppose ℓ ∤ pDN is any prime such that Frob_{ℓ}(K(E[p])/Q) = [τ]. By Lemma 17.6.4, [K[ℓ] : H] = ℓ+1 is divisible by p, so there is a unique extension H′ of H of degree p contained in K[ℓ]. Let φ be a lift of Frob_{ℓ}(H/Q) to Gal(H′/Q), and let

z′ = Tr_{K[ℓ]/H′}(y_{ℓ} − φ(y_{ℓ})) − \frac{a_{ℓ}}{p} (y_{H} − φ(y_{H})) ∈ E(H′).

Lemma 17.6.6. Tr_{H′/H}(z′) = 0.

Proof. Using properties of the trace and Proposition 17.6.5, we have

Tr_{H′/H}(z′) = Tr_{K[ℓ]/H}(y_{ℓ}) − Tr_{K[ℓ]/H}(φ(y_{ℓ})) − a_{ℓ}(y_{H}) − a_{ℓ}φ(y_{H})
= a_{ℓ}y_{H} − a_{ℓ}φ(y_{H}) − a_{ℓ}(y_{H}) − a_{ℓ}φ(y_{H})
= 0.

For the rest of the proof we only consider primes p such that

H^{1}(Q_{v}^{unr}/Q_{v}, E(Q_{v}^{unr}))[p] = 0

for all v, where Q_{v}^{unr} is the maximal unramified extension of Q_{v}. (This only excludes finitely many primes, by [Milne, Arithmetic Duality Theorems, Prop. I.3.8].)

Proposition 17.6.7. There is an element c_{ℓ} ∈ H^{1}(Q, E)[p] such that res_{v}(c_{ℓ}) = 0 for all v ≠ ℓ, and res_{ℓ}(c_{ℓ}) ≠ 0 if and only if y ∉ pE(K_{ℓ}).

Proof. Recall that we assumed that p ∤ [H : K]. Thus there is a unique extension K′ of K of degree p in K[ℓ], which is totally ramified at ℓ and unramified everywhere else, such that H′ = HK′. Let

z = Tr_{H′/K′}(z′) ∈ E(K′).

By Lemma 17.6.6, Tr_{K′/K}(z) = 0.

The extension Gal(K′/K) is cyclic (since it has degree p), so it is generated by a single element, say σ. The cohomology of a cyclic group is easy to understand (see Atiyah-Wall^{10} ). In particular, we have a canonical isomorphism 10

H^{1}(K′/K, E(K′)) ≅ \frac{ker(Tr_{K′/K} : E(K′) → E(K))}{(σ − 1)E(K′)}.

Define c′_{ℓ} to be the element of H^{1}(K′/K, E(K′)) that corresponds to z under this isomorphism. A more careful analysis shows that c′_{ℓ} is the restriction of an element c_{ℓ} in H^{1}(Q, E)[p].

If v ≠ ℓ then

res_{v}(c_{ℓ}) ∈ H^{1}(Q_{v}^{unr}/Q_{v}, E(Q_{v}^{unr}))[p] = 0,

as asserted. The assertion about res_{ℓ}(c_{ℓ}) is obtained with some work by reducing modulo ℓ and applying Lemma 17.6.4 and Proposition 17.6.5.

Remark 17.6.8. McCallum has observed that c_{ℓ} is represented by the 1-cocycle

σ ↦ −\frac{(σ − 1)y_{ℓ}}{ℓ} ,

where ((σ−1)y_{ℓ})/ℓ is the unique point P ∈ E(K[ℓ]) such that ℓP = (σ−1)y_{ℓ}. (The point P is unique because no nontrivial p-torsion on E is defined over E(K[ℓ]).)

10expand

Corollary 17.6.9. If y ∉ pE(K_{ℓ}), then res_{ℓ}(Sel^{(p)}(E)) = 0.

Proof. Combine Propositions 17.6.2 and 17.6.7.

If t ∈ H^{1}(K, E[p]), denote by t̂ the image of t under restriction:

H^{1}(K, E[p]) → Hom(Gal(\overline{Q}/K(E[p])), E[p]).  (17.6.1)

It is useful to consider t̂, since homomorphisms satisfy a local-to-global principle. A homomorphism φ : Gal(\overline{Q}/F) → M is 0 if and only if it is 0 when restricted to Gal(\overline{F}_{λ}/F_{λ}) for all primes λ of F (this fact follows from the Chebotarev density theorem).

For the rest of the proof we now assume that p is large enough that there are no Q-rational cyclic subgrups of E of order p, and H^{1}(K(E[p])/K, E[p]) = 0. (That we can do this follows from a theorem of Serre when E does not have CM, or CM theory when E does have CM.)

Lemma 17.6.10. Suppose t ∈ H^{1}(K, E[p])^{±} and the image of t̂ is cyclic. Then t = 0.

Proof. Since τ acts on t̂ by ±, the image of t̂ is rational over Q. Thus the image of τ is trivial. The kernel of the restriction map (17.6.1) is also trivial by assumption, so t = 0.

We are now ready to prove Theorem 17.6.1. Choose p large enough so that y ∉ pE(K), in addition to all other “sufficiently large” constraints that we put on p above.

Fix s ∈ Sel^{(p)}(E) and write ŝ for the restriction of s to a homomorphism Gal(\overline{Q}/K(E[p])) → E[p]. Our goal is to prove that s = 0. Write ŷ for the restriction to Gal(\overline{Q}/K(E[p])) of the image of y under the injection

E(K)^{−}/pE(K)^{−} → H^{1}(K, E[p])^{−}.  (17.6.2)

Fix a finite extension F of K(E[p]) that is Galois over Q, so that both homomorphism ŝ and ŷ factor through G = Gal(F/K(E[p])).

Suppose γ ∈ G, and use the Chebotarev Density Theorem to find a prime ℓ ∤ pDN such that Frob_{ℓ}(F/Q) = [γτ]. Then

Frob_{ℓ}(K(E[p])/Q) = [τ],

and

Frob_{ℓ}(F/K(E[p])) = [γτ]^{order(τ)} = [(γτ)^{2}].

By Lemma 17.6.10, s_{ℓ} = 0 implies that ŝ((γτ)^{2}) = 0 for all γ ∈ G. Likewise, y ∈ pE(K_{ℓ}) implies that ŷ((γτ)^{2}) = 0 for all γ ∈ G. Since s ∈ H^{1}(Q, E) we have τ(ŝ) = ŝ and by (17.6.2) we have τ(ŷ) = −ŷ, so

ŝ((γτ)^{2}) = ŝ(γ) + ŝ(τγτ) = (1 + τ)ŝ(γ)  (17.6.3)
ŷ((γτ)^{2}) = ŷ(γ) + ŷ(τγτ) = (1 − τ)ŷ(γ).  (17.6.4)

By Corollary 17.6.9, for every γ ∈ G, one of ŝ((γτ)^{2}) or ŷ((γτ)^{2}) is 0, so by (17.6.3) and (17.6.4) at least one of the following holds:

ŝ(γ) ∈ E[p]^{−}

or

ŷ(γ) ∈ E[p]^{+}.

Thus

G = ŝ^{−1}(E[p]^{−}) ⋃ ŷ^{−1}(E[p]^{+}).

Since a group cannot be the union of two proper subgroups, either ŝ(G) ⊂ E[p]^{−} or ŷ(G) ⊂ E[p]^{+}. By Lemma 17.6.10, either s = 0 or y ∈ pE(K). But we assumed that y ∉ pE(K), so s = 0, as claimed.

17.6.1 A Heegner point when N = 11

Let E be the elliptic curve X_{0}(11), which has conductor 11 and Weierstrass equation y^{2} + y = x^{3} − x^{2} − 10x − 20. The following uses a nonstandard MAGMA package...

> E := EllipticCurve(CremonaDatabase(),"11A");
> E;
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
>
[]
> A := [-D : D in [1..20] | HeegnerHypothesis(E,-D)]; A;
[ -7, -8, -19 ]
> P := [* *]; for D in A do time p,q := HeegnerPoint(E,D); Append(~P,<p,q>); end for;
> P;   // pairs the (heegner point, its trace to Q)
[* <(0.500000000000000000023668707008012648357 + 1.322875655532295292727422603610*i :
    -2.0000000000000000043473199962953622702 + 5.291502622129181186268103643387*i : 1),
    (11 : -61 : 1)>,
<(-2.99999999999999999717372417577 + 1.414213562373095051128215430668*i :
    2.99999999999999999372415320819 + 4.242640687119285158529184817212*i : 1),
     (-3/4 : 121/8 : 1)>,
<(-6.99999999999999999753082167926 + 8.71779788708134709829181547345*i :
   37.000000000000000044867139452136100922 + 8.71779788708134710182445372185*i : 1),
   (1161/16 : -33549/64 : 1)> *]
> P[1];
<(0.500000000000000000023668707008012648357 + 1.322875655532295292727422603610*i :
   -2.0000000000000000043473199962953622702 + 5.291502622129181186268103643387*i : 1),
  (11 : -61 : 1)>
> Sqrt(-7)/2;
2.64575131106459059050161575359*i
> Sqrt(-7)/2;
1.322875655532295295250807876795*i
> 2*Sqrt(-7);
5.291502622129181181003231507183*i
> // Thus  ((1+sqrt(-7))/2 , -2+2*sqrt(-7)) is the Heegner point
> // corresponding to Q(sqrt(-7)).

17.6 Kolyvagin’s Euler system of Heegner points 177

17.6.2 Kolyvagin’s Euler system for curves of rank at least 2

11

11

11Summary of my work to show nontriviality in some cases. Make sure to also mention work of Rubin, Mazur, Howard, Weinstein, etc., in this section. One interesting idea might be to add my whole book on BSD into this book...

178 17. The Birch and Swinnerton-Dyer Conjecture

The Gorenstein Property for Hecke Algebras

18.1 Mod ℓ representations associated to modular forms

Suppose f = ∑ a_{n}q^{n} is a newform of exact level N and weight 2 for the congruence subgroup Γ_{0}(N). Let E = Q(…, a_{n}, …) and let λ be a place of E lying over the prime ℓ of Q. The action of Gal(\overline{Q}/Q) on the ℓ-adic Tate module of the associated abelian variety A_{f} gives rise to a representation

ρ_{λ} : Gal(\overline{Q}/Q) → GL_{2}(E_{λ})

that satisfies det(ρ_{λ}) = χ_{ℓ} and tr(ρ_{λ}(Frob_{p})) = a_{p} for p ∤ ℓN. Using the following lemma, it is possible to reduce ρ_{λ} module λ.

Lemma 18.1.1. Let O be the ring of integers of E_{λ}. Then ρ_{λ} is equivalent to a representation that takes values in GL_{2}(O).

Proof. View GL_{2}(E_{λ}) as the group of automorphisms of a 2-dimensional E_{λ}-vector space V. A lattice L ⊂ V is a free O-module of rank 2 such that L ⊗ E_{λ} ≅ V. It suffices to find an O-lattice L in V that is invariant under the action via ρ_{λ} of Gal(\overline{Q}/Q). For then the matrices of Gal(\overline{Q}/Q) with respect to a basis of V consisting of vectors from L will have coefficients in O. Choose any lattice L_{0} ⊂ V. Since L_{0} is discrete and Gal(\overline{Q}/Q) is compact, the set of lattices ρ_{λ}(g)L_{0} with g ∈ Gal(\overline{Q}/Q) is finite. Let L = ∑ ρ_{λ}(g)L_{0} be the sum of the finitely many conjugates of L_{0}; then L is Galois invariant. The sum is a lattice because it is finitely generated and torsion free, and O is a principal ideal domain.

Choose an O as in the lemma, and tentatively write \overline{ρ}_{λ} = ρ_{λ} mod λ:

\xymatrix{Gal(\overline{Q}/Q) \ar[dr]_{_{false \overline{ρ}_{λ}}} \ar[r]^{ρ_{λ}} & GL_{2}(O) \ar[d] \\  & GL_{2}(O/λ)}

The drawback to this definition is that \overline{ρ}_{λ} is not intrinsic; the definition depends on making a choice of O. Instead we define \overline{ρ}_{λ} to be the semisimplification of the reduction of ρ_{λ} modulo λ. The semisimplification of a representation is the direct sum of the Jordan-Hölder factors in a filtration of a vector space affording the representation. We have det(\overline{ρ}_{λ}) ≡ χ_{ℓ} (mod ℓ), where χ_{ℓ} is the mod ℓ cyclotomic character, and tr(\overline{ρ}_{λ}(Frob_{p})) = a_{p} mod λ. Thus the characteristic polynomials in the semisimplification \overline{ρ}_{λ} are independent of our choice of reduction of ρ_{λ}. The following theorem implies that \overline{ρ}_{λ} depends only on f and λ, and not on the choice of the reduction.

Theorem 18.1.2 (Brauer-Nesbitt). Suppose ρ_{1}, ρ_{2} : G → GL(V) are two finite dimensional semisimple representations of a group G over a finite field k. Assume furthermore that for every g ∈ G the characteristic polynomial of ρ_{1}(g) is the same as the characteristic polynomial of ρ_{2}(g). Then ρ_{1} and ρ_{2} are equivalent.

Proof. For a proof, see [CR62, §30, p. 215].

The Hecke operators T_{n} act as endomorphisms of S_{2}(Γ_{0}(N)). Let

T := Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N))),

and recall that T is a commutative ring; as a Z-module T has rank equal to dim_{C} S_{2}(Γ_{0}(N)). Let 𝔪 be a maximal ideal of T and set k := T/𝔪 ≈ F_{ℓ^{ν}}.

Proposition 18.1.3. There is a unique semisimple representation

ρ_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(k)

such that ρ_{𝔪} is unramified outside ℓN and

tr(ρ_{𝔪}(Frob_{p})) = T_{p} mod 𝔪
det(ρ_{𝔪}(Frob_{p})) = p mod 𝔪.

Proof. It is enough to prove the assertion with T replaced by the subalgebra

T_{0} = Z[{…, T_{n}, …: (n, N) = 1}].

Indeed, the maximal ideal 𝔪 of T pulls back to a maximal ideal 𝔪_{0} of T_{0}, and k_{0} = T_{0}/𝔪_{0} ⊂ k. Now

T_{0} ⊂ T_{0} ⊗ Q = ∏_{i=1}^{t} E_{i}

with the E_{i} number fields. Let O_{E_{i}} be the ring of integers of E_{i} and let O = ∏ O_{E_{i}}. By the going up theorem there is a maximal ideal λ ⊂ ∏ O_{E_{i}} lying over 𝔪_{0}:

[figure: λ; ∏ O_{E_{i}}; O/λ; 𝔪_{0}; T_{0}; k_{0}]

Using the above construction, we make a representation

\overline{ρ}_{λ} : Gal(\overline{Q}/Q) → GL_{2}(O/λ)

18.1 Mod ℓ representations associated to modular forms 181

that satisfies det(\overline{ρ}_{λ}) ≡ χ_{ℓ} (mod λ) and tr(\overline{ρ}_{λ}) = T_{p} (mod λ). Because of how we have set things up, T_{p} plays the role of a_{p}. Thus this representation has the required properties, but it takes values in GL_{2}(O/λ) instead of k_{0}.

Since the characteristic polynomial of every \overline{ρ}_{λ}(g) for g ∈ Gal(\overline{Q}/Q) has coefficients in the subfield k_{0} ⊂ O/λ there is a model for ρ_{λ} over k_{0}. This is a classical result of I. Schur. Brauer groups of finite fields are trivial (see e.g., [Ser79, X.7, Ex. a]), so the argument of [Ser77, §12.2] completes the proof of the proposition.

Alternatively, when the residue characteristic ℓ of k_{0} is odd, the following more direct proof can be used. Complex conjugation acts through \overline{ρ}_{λ} as a matrix with distinct F_{ℓ}-rational eigenvalues; another well known theorem of Schur [Sch06, IX a] (cf. [Wal85, Lemme I.1]) then implies that \overline{ρ}_{λ} can be conjugated into a representation with values in GL(2, k_{0}).

Let us look at this construction in another way. Write

T_{0} ⊗ Q = E_{1} × ⋯ × E_{t}

and recall that each number field E_{i} corresponds to a newform of level M | N; one can obtain E_{i} by adjoining the coefficients of some newform of level M | N to Q. Likewise, the Jacobian J_{0}(N) is isogenous to a product A_{1} ×⋯×A_{t}. Consider one of the factors, say E_{1}, and to fix ideas suppose that it corresponds to a newform of exact level N. Since Tate_{ℓ} A_{1} is free of rank 2 over E_{1} ⊗_{Q} Q_{ℓ}, we obtain a 2-dimensional representation ρ_{λ}. Reducing mod λ and semisimplifying gives the representation constructed in the above proposition. But it is also possible that one of the fields E_{i} corresponds to a newform f of level M properly dividing N. In this case, we repeat the whole construction with J_{0}(M) to get a 2-dimensional representation.

Consider one of the abelian varieties A_{1} as above, which we view as an abelian subvariety of J_{0}(N). Then T acts on A_{1}; let \overline{T} denote the image of T in End A_{1}. Although \overline{T} sits naturally in O_{1}, which is the ring of integers of a field, \overline{T} might not be integrally closed. Consider the usual Z_{ℓ}-adic Tate module Tate_{ℓ}(A_{1}) ≈ Z_{ℓ}^{2 dim A_{1}}, where dim A_{1} = [E_{1} : Q]. In the 1940s Weil proved that \overline{T} ⊗ Z_{ℓ} acts faithfully on the Tate module. By the theory of semilocal rings (see, e.g., [Eis95, Cor. 7.6]), we have

\overline{T} ⊗ Z_{ℓ} = ∏_{𝔪|ℓ} \overline{T}_{𝔪},

where the product is over all maximal ideals of \overline{T} of residue characteristic ℓ. The idempotents e_{𝔪}, in this decomposition, decompose Tate_{ℓ} as a product ∏_{𝔪} Tate_{𝔪}(A_{1}). It would be nice if Tate_{ℓ} were free of rank 2 over \overline{T} ⊗ Z_{ℓ} but this is not known to be true in general, although it has been verified in many special cases. For this to be true we must have that, for all 𝔪 | ℓ, that Tate_{𝔪}(A_{1}) is free of rank 2 over \overline{T}_{𝔪}.

Next we put things in a finite context instead of a projective limit context. Let J = J_{0}(N), then by Albanese or Picard functoriality T ⊂ End J. Let 𝔪 ⊂ T be a maximal ideal. Let

J[𝔪] = \{t ∈ J(\overline{Q}) : xt = 0 for all x ∈ 𝔪\}.

Note that J[𝔪] ⊂ J[ℓ] where ℓ is the rational prime lying in 𝔪. Now J[ℓ] is an F_{ℓ} vector space of rank 2g where g is the genus of X_{0}(N). Although it is true that J[ℓ] is a T/ℓ-module, it is not convenient to work with T/ℓ since it might not be a product of fields because of unpleasant ramification. It is more convenient to work with J[𝔪] since T/𝔪 is a field. Thus, via this optic, J[𝔪] is a k[G]-module where k = T/𝔪 and G = Gal(\overline{Q}/Q). The naive hope is that J[𝔪] is a model for ρ_{𝔪}, at least when ρ_{𝔪} is irreducible. This does not quite work, but we do have the following theorem.

Theorem 18.1.4. If ℓ ∤ 2N then J[𝔪] is a model for ρ_{𝔪}.

18.2 The Gorenstein property

Consider the Hecke algebra

T := Z[…, T_{n}, …] ⊂ End J_{0}(N),

and let 𝔪 ⊂ T be a maximal ideal of residue characteristic ℓ. We have constructed a semisimple representation

ρ_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(T/𝔪).

It is unramified outside ℓN, and for any prime p ∤ ℓN we have

tr(ρ_{𝔪}(Frob_{p})) = T_{p} mod 𝔪
det(ρ_{𝔪}(Frob_{p})) = p mod 𝔪.

We will usually be interested in the case when ρ_{𝔪} is irreducible. Let T_{𝔪} = \underleftarrow{lim} T/𝔪^{i}T denote the completion of T at 𝔪. Note that T ⊗_{Z} Z_{ℓ} = ∏_{𝔪|ℓ} T_{𝔪}. Our goal is to prove that if 𝔪 ∤ 2N and ρ_{𝔪} is irreducible, then T_{𝔪} is Gorenstein.

Definition 18.2.1. Let O be a complete discrete valuation ring. Let T be a local O-algebra which as a module is finite and free over O. Then T is a Gorenstein O-algebra if there is an isomorphism of T -modules T ⟶^{∼} Hom_{O}(T, O).

Thus T_{𝔪} is Gorenstein if there is an isomorphism Hom_{Z_{ℓ}}(T_{𝔪}, Z_{ℓ}) ≈ T_{𝔪} of T_{𝔪}-modules. Intuitively, this means that T_{𝔪} is “autodual”.

Theorem 18.2.2. Let J = J_{0}(N) and let 𝔪 be a maximal ideal of the Hecke algebra. Assume that ρ_{𝔪} is irreducible and that 𝔪 ∤ 2N. Then dim_{T/𝔪} J[𝔪] = 2 and J[𝔪], as a Galois module, is a 2-dimensional representation giving rise to ρ_{𝔪}.

An easy argument shows that the theorem implies T_{𝔪} is Gorenstein.

We first consider the structure of W = J[𝔪]. Suppose the two dimensional representation

ρ_{𝔪} : Gal(\overline{Q}/Q) → Aut_{T/𝔪} V

constructed before is irreducible. Consider the semisimplification W^{ß} of W, thus W^{ß} is the direct sum of its Jordan-Hölder factors as a Gal(\overline{Q}/Q)-module. Mazur proved the following theorem.

Theorem 18.2.3. There is some integer t ≥ 0 so that

W^{ß} ≅ V × ⋯ × V = V^{t}.

If ρ_{𝔪} is in fact absolutely irreducible then it is a result of Boston, Lenstra, and Ribet [BpR91] that W ≅ V × ⋯ × V. A representation is absolutely irreducible if it is irreducible over the algebraic closure. It can be shown that if ℓ ≠ 2 and ρ_{𝔪} is irreducible then ρ_{𝔪} must be absolutely irreducible.

The construction of W is nice and gentle whereas the construction of V is accomplished via brute force.

Proof. (Mazur) We want to compare V with W. Let d = dim W. Let

W^{∗} = Hom_{T/𝔪}(W, T/𝔪(1))

where T/𝔪(1) = T/𝔪 ⊗_{Z} μ_{ℓ}. We need to show that

W^{ß} ⊕ W^{∗ß} ≅ V × ⋯ × V = V^{d}

as representations of Gal(\overline{Q}/Q). Note that each side is a semisimple module of dimension 2d. To obtain the isomorphism we show that the two representations have the same characteristic polynomials so they are isomorphic.

We want to show that the characteristic polynomial of Frob_{p} is the same for both W^{ß} ⊕ W^{ß} and V × ⋯ × V. The characteristic polynomial of Frob_{p} on V is X^{2} − T_{p}X + p = (X − r)(X − pr^{−1}) where r lies in a suitable algebraic closure. It follows that the characteristic polynomial of Frob_{p} on V^{d} is (x − r)^{d}(x − pr^{−1})^{d}. On W the characteristic polynomial of Frob_{p} is (X − α_{1}) ⋯ (X − α_{d}) where α_{i} is either r or pr^{−1}. This is because Eichler-Shimura implies Frob_{p} must satisfy Frob_{p}^{2} −T_{p} Frob_{p} +p = 0. [[I don’t see this implication.]] On W^{∗} the characteristic polynomial of Frob_{p} is (X − pα_{1}^{−1}) ⋯ (X − pα_{d}^{−1}). [[This is somehow tied up with the definition of W^{∗} and I can’t quite understand it.]] Thus on W ⊕ W^{∗}, the characteristic polynomial of Frob_{p} is

∏_{i=1}^{d}(X − α_{i})(X − pα_{i}^{−1}) = ∏_{i=1}^{d}(X − r)(X − pr^{−1}) = (X − r)^{d}(X − pr^{−1})^{d}.

Therefore the characteristic polynomial of Frob_{p} on W ⊕ W^{∗} is the same as the characteristic polynomial of Frob_{p} on V × ⋯ × V. The point is that although the α_{i} could all a priori be r or pr^{−1}, by adding in W^{∗} everything pairs off correctly. [[I don’t understand why we only have to check that the two representations agree on Frob_{p}. There are lots of other elements in Gal(\overline{Q}/Q), right?]]

We next show that J[𝔪] ≠ 0. This does not follow from the theorem proved above because it does not rule out the possibility that t = 0 and hence W^{ß} ≅ 0. Suppose J[𝔪] = 0, then we will show that J[𝔪^{i}] = 0 for all i ≥ 1. We consider the ℓ-divisible group J_{𝔪} = ∪_{i}J[𝔪^{i}]. To get a better feel for what is going on, temporarily forget about 𝔪 and just consider the Tate module corresponding to ℓ.

It is standard to consider the Tate module

Tate_{ℓ} J = \underleftarrow{lim} J[ℓ^{i}] ≅ Z_{ℓ}^{2 dim J}.

It is completely equivalent to consider

J_{ℓ} := ∪_{i=1}^{∞}J[ℓ^{i}] ⟶^{∼} (Q_{ℓ}/Z_{ℓ})^{2 dim J}.

Note that since Q_{ℓ}/Z_{ℓ} is not a ring the last isomorphism must be viewed as an isomorphism of abelian groups. In [Maz77] Mazur called Tate_{ℓ} J ≅ Hom(Q_{ℓ}/Z_{ℓ}, J_{ℓ}) the covariant Tate module. Call

Tate_{ℓ}^{∗} J := Hom(J_{ℓ}, Q_{ℓ}/Z_{ℓ}) ≅ Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ})

the contravariant Tate module. [[Why are the last two isomorphic?]] The covariant and contravariant Tate modules are related by a Weil pairing J[ℓ^{i}] × J[ℓ^{i}] → μ_{ℓ^{i}}. Taking projective limits we obtain a pairing

⟨·, ·⟩ : Tate_{ℓ} J × Tate_{ℓ} J → Z_{ℓ}(1) = \underleftarrow{lim} μ_{ℓ^{i}}.

This gives a map

Tate_{ℓ} J → Hom(Tate_{ℓ} J, Z_{ℓ}(1)) = (Tate_{ℓ}^{∗} J)(1)

where (Tate_{ℓ}^{∗} J)(1) = (Tate_{ℓ}^{∗} J) ⊗ Z_{ℓ}(1). Z_{ℓ}(1) is a Z_{ℓ}-module where

∑ a_{i}ℓ^{i} · ζ = ζ^{∑ a_{i}ℓ^{i}}

[[This should probably be said long ago.]] This pairing is not a pairing of Tmodules, since if t ∈ T then ⟨tx, y⟩ = ⟨x, t^{∨}y⟩. It is more convenient to use an adapted pairing defined as follows. Let w = w_{V} ∈ End J_{0}(N) be the Atkin-Lehner involution so that t^{∨} = wtw. Define a new T-compatible pairing by [x, y] := ⟨x, wy⟩. Then

[tx, y] = ⟨tx, wy⟩ = ⟨x, t^{∨}wy⟩ = ⟨x, wty⟩ = [x, ty].

The pairing [·, ·] defines an isomorphism of T ⊗ Z_{ℓ}-modules

Tate_{ℓ} J ⟶^{∼} Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ}(1)).

Since Z_{ℓ}(1) is a free module of rank 1 over Z_{ℓ} a suitable choice of basis gives an isomorphism of T ⊗ Z_{ℓ}-modules

(Tate_{ℓ}^{∗} J)(1) ≅ Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ}(1)) ≅ Hom_{Z_{ℓ}}(Tate_{ℓ} J, Z_{ℓ}) = Tate_{ℓ}^{∗} J.

Thus Tate_{ℓ} J ⟶^{∼} Tate_{ℓ}^{∗} J.

Proof. (That J[𝔪] ≠ 0.) The point is that the contravariant Tate module Hom(J_{ℓ}, Q_{ℓ}/Z_{ℓ}) is the Pontrjagin dual of T_{ℓ}. How does this relate to Tate_{𝔪} J? Since T ⊗ Z_{ℓ} ≅ ∏_{𝔪|ℓ} T_{𝔪}, Tate_{ℓ} J ≅ ∏_{𝔪|ℓ} Tate_{𝔪} J so we can define Tate_{𝔪}^{∗} J := Hom_{Z_{ℓ}}(Tate_{𝔪} J, Z_{ℓ}). Weil proved that Tate_{𝔪} J ≅ T ate_{𝔪}^{∗}J is nonzero. View Tate_{𝔪}^{∗} J as being dual to J_{𝔪} in the sense of Pontrjagin duality and so (Tate_{𝔪}^{∗} J)/(𝔪 Tate_{𝔪}^{∗} J) is dual to J[𝔪]. If J[𝔪] = 0 then this quotient is 0, so Nakayama’s lemma would imply that Tate_{𝔪}^{∗} J = 0. This would contradicts Weil’s assertion. Therefore J[𝔪] ≠ 0.

18.3 Proof of the Gorenstein property

We are considering the situation with respect to J_{0}(N) although we could consider J_{1}(N). Let T ⊂ End J_{0}(N) be the Hecke algebra and let 𝔪 ⊂ T be a maximal ideal.

Let ℓ be the characteristic of the residue class field T/𝔪. Let T_{𝔪} = \underleftarrow{lim} T/𝔪^{i}T. Then T ⊗_{Z} Z_{ℓ} = ∏_{𝔪|ℓ} T_{𝔪}. [[I want to put a good reference for this Atiyah-Macdonald like fact here.]] Each T_{𝔪} acts on Tate_{ℓ} J_{0}(N) so we obtain a product decomposition

Tate_{ℓ} J_{0}(N) = ∏_{𝔪|ℓ} Tate_{𝔪} J_{0}(N).

We have the following two facts:

1. Tate_{𝔪} J_{0}(N) ≠ 0

2. Tate_{ℓ} J_{0}(N) is T ⊗ Z_{ℓ}-autodual and each Tate_{𝔪} J_{0}(N) is Z_{ℓ}-autodual.

[[autoduality for which dual? I think it is the linear dual since this is used.]]

Let W = J[𝔪], then the action of Gal(\overline{Q}/Q) on W gives a representation of Gal(\overline{Q}/Q) over the field T/𝔪. We compared W with a certain two dimensional representation ρ_{𝔪} : Gal(\overline{Q}/Q) → V over T/𝔪. Assume unless otherwise stated that V is irreducible as a Gal(\overline{Q}/Q)-module. Let Tate_{ℓ} = Tate_{ℓ} J_{0}(N) and Tate_{𝔪} = Tate_{𝔪} J_{0}(N). A formal argument due to Mazur showed that

W^{ß} ≅ V × ⋯ × V = V^{⊕t}.

We have not yet determined t but we would like to show that t = 1.

Definition 18.3.1. The Pontrjagin dual of a module M is the module M^{∧} := Hom(M, Q/Z) where M is viewed as an abelian group (if M is topological, only take those homomorphisms whose kernel is compact). The linear dual of a module M over a ring R is the module M^{∗} = Hom_{R}(M, R).

Exercise 18.3.2. Note that (Q_{ℓ}/Z_{ℓ})^{∧} = Z_{ℓ} and Z_{ℓ}^{∧} = Q_{ℓ}/Z_{ℓ}.

Solution.. We can think of Q_{ℓ}/Z_{ℓ} as

{ ∑_{n=−k}^{−1} a_{n}ℓ^{n} : k > 0 and 0 ≤ a_{i} < ℓ}.

Let (b_{i}) ∈ Z_{ℓ} so b_{i} ∈ Z/ℓ^{i}Z and b_{i+1} ≡ b_{i} (mod ℓ^{i}). Define a map Q_{ℓ}/Z_{ℓ} → Q/Z by 1/ℓ^{i} ↦ b_{i}/ℓ^{i}. To check that this is well-defined it suffices to check that 1/ℓ^{i} maps to the same place as ℓ · 1/ℓ^{i+1}. Now 1/ℓ^{i} ↦ b_{i}/ℓ^{i} and

ℓ · 1/ℓ^{i+1} ↦ ℓ · b_{i+1}/ℓ^{i+1} = b_{i+1}/ℓ^{i}.

So we just need to check that

b_{i+1}/ℓ^{i} ≡ b_{i}/ℓ^{i} (mod Z).

This is just the assertion that (b_{i+1} − b_{i})/ℓ^{i} ∈ Z which is true since b_{i+1} ≡ b_{i} (mod ℓ^{i}).

Proposition 18.3.3. Let the notation be as above, then t > 0.

Proof. The idea is to use Nakayama’s lemma to show that if t = 0 and hence W = 0 then Tate_{𝔪} = 0 which is clearly false. But the relation between W and Tate_{𝔪} is rather convoluted. In fact J[ℓ^{∞}] is the Pontrjagin dual of Tate_{ℓ}^{∗}, that is,

J[ℓ^{∞}]^{∧} = Tate_{ℓ}^{∗} = Hom_{Z_{ℓ}}(Tate_{ℓ}, Z_{ℓ})

and

(Tate_{ℓ}^{∗})^{∧} = Hom(Tate_{ℓ}^{∗}, Q_{ℓ}/Z_{ℓ}) = J[ℓ^{∞}].

[[First: Why are they dual? Second: Why are we homing into Q_{ℓ}/Z_{ℓ} instead of Q/Z?]] Looking at the 𝔪-adic part shows that

J[𝔪^{∞}] = Hom(Tate_{𝔪}^{∗}, Q_{ℓ}/Z_{ℓ})

and hence

J[𝔪] = Hom(Tate_{𝔪}^{∗} /𝔪 Tate_{𝔪}^{∗}, Z/ℓZ).

Thus if J[𝔪] = 0 then Nakayama’s lemma implies Tate_{𝔪}^{∗} = 0. By autoduality this implies Tate_{𝔪} = 0.

We have two goals. The first is to show that t = 1, i.e., that J[𝔪] is 2-dimensional over T/𝔪. The second is to prove that T_{𝔪} is Gorenstein, i.e., that T_{𝔪} ≅ Hom_{Z_{ℓ}}(T_{𝔪}, Z_{ℓ}). This is one of the main theorems in the subject. We are assuming throughout that ρ_{𝔪} is irreducible and ℓ ∤ 2N. Loosely speaking the condition that ℓ |̸2N means that J[𝔪] has good reduction at ℓ and that J[𝔪] can be understood just by understanding J[𝔪] in characteristic ℓ. We want to prove that T_{𝔪} is Gorenstein because this property plays an essential role in proving that T_{𝔪} is a local complete intersection. Example 18.3.4. Let

T = {(a, b, c, d) ∈ Z_{p}^{4} : a ≡ b ≡ c ≡ d (mod p)}.

Then T is a local ring that is not Gorenstein.

For now we temporarily postpone the proof of the first goal and instead show that the first goal implies the second.

Theorem 18.3.5. Suppose J[𝔪] is two dimensional over T/𝔪 (thus t = 1). Then T_{𝔪} is Gorenstein.

Proof. We have seen before that

J[𝔪] = Hom_{Z/ℓZ}(Tate_{𝔪}^{∗} /𝔪 Tate_{𝔪}^{∗}, Z/ℓZ)
= Hom_{T/𝔪}(Tate_{𝔪}^{∗} /𝔪 Tate_{𝔪}^{∗}, T/𝔪).

Thus the dual of Tate_{𝔪}^{∗} /𝔪 Tate_{𝔪}^{∗} is two dimensional over T/𝔪 and hence Tate_{𝔪}^{∗} /𝔪 Tate_{𝔪}^{∗} itself is two dimensional over T/𝔪. By Nakayama’s lemma and autoduality of Tate_{𝔪} this implies Tate_{𝔪} is generated by 2 elements over T_{𝔪}. There is a surjection

T_{𝔪} × T_{𝔪} → → Tate_{𝔪} .

In fact it is true that rank_{Z_{ℓ}} Tate_{𝔪} = 2 rank_{Z_{ℓ}} T_{𝔪}. We temporarily postpone the proof of this claim. Assuming this claim and using that a surjection between Z_{ℓ}- modules of the same rank is an isomorphism implies that Tate_{𝔪} ≅ T_{𝔪} × T_{𝔪}. Now T_{𝔪} is a direct summand of the free Z_{ℓ} module Tate_{𝔪} so T_{𝔪} is projective. A projective module over a local ring is free. Thus T_{𝔪} is free of rank 1 and hence autodual (Gorenstein). [[This argument is an alternative to Mazur’s – it seems too easy... maybe I am missing something.]]

We return to the claim that

rank_{Z_{ℓ}} Tate_{𝔪} = 2 rank_{Z_{ℓ}} T_{𝔪}.

This is equivalent to the assertion that

dim_{Q_{ℓ}} Tate_{𝔪} ⊗_{Z_{ℓ}}Q_{ℓ} = 2 dim_{Q_{ℓ}} T_{𝔪} ⊗_{Z_{ℓ}} Q_{ℓ}.

The module Tate_{ℓ} J_{0}(N) is the projective limit of the ℓ-power torsion on the Jacobian

J(C) = \frac{Hom_{C}(S_{2}(Γ_{0}(N), C), C)}{H_{1}(X_{0}(N), Z)}.

Let L = H_{1}(X_{0}(N), Z) be the lattice. Then L is a T-module and Tate_{ℓ} = L ⊗_{Z} Z_{ℓ} (since L/ℓ^{i}L ≅ (\frac{1}{ℓ^{i}} L)/L). Tensoring with R gives

L ⊗_{Z} R = Hom_{C}(S_{2}(Γ_{0}(N), C), C)
= Hom_{R}(S_{2}(Γ_{0}(N), R), C)
= Hom_{R}(S_{2}(Γ_{0}(N), R), R) ⊗_{R} C = (T ⊗_{Z} R) ⊗_{R} C

Thus L⊗_{Z}R is free of rank 2 over T⊗_{Z}R and L⊗_{Z}C is free of rank 2 over T⊗_{Z}C. Next choose an embedding Q_{ℓ} ↪ C. Now Tate_{ℓ} is a module over T⊗Z_{ℓ} = ∏_{𝔪|ℓ} T_{𝔪} so we have a decomposition Tate_{ℓ} = ∏_{𝔪|ℓ} Tate_{𝔪}. Since

Tate_{ℓ} ⊗_{Z_{ℓ}}Q_{ℓ} = ∏_{𝔪} Tate_{𝔪} ⊗_{Z_{ℓ}}Q_{ℓ}

we can tensor with C to see that

Tate_{ℓ} ⊗_{Z_{ℓ}}C = ∏_{𝔪} Tate_{𝔪} ⊗_{Z_{ℓ}}C.

But Tate_{ℓ} ⊗_{Z_{ℓ}}C = L ⊗_{Z} C is free of rank 2 over T ⊗ C. Therefore the product ∏_{𝔪} Tate_{𝔪} ⊗_{Z_{ℓ}}C is free of rank 2 over T ⊗ C. Since

T ⊗ C = (T ⊗_{Z} Z_{ℓ}) ⊗_{Z_{ℓ}} C = ∏_{𝔪}(T_{𝔪} ⊗_{Z_{ℓ}} C)

we conclude that for each 𝔪, Tate_{𝔪} ⊗_{Z_{ℓ}}C is free of rank 2 over T_{𝔪} ⊗_{Z_{ℓ}} C. This implies

dim_{C} Tate_{𝔪} ⊗_{Z_{ℓ}}C = 2 dim_{C} T_{𝔪} ⊗_{Z_{ℓ}} C

which completes the proof.

18.3.1 Vague comments

Ogus commented that this same proof shows that T ⊗_{Z} C is Gorenstein. Then he said that something called “faithfully flat descent” could then show that T ⊗_{Z} Q is Gorenstein.

We have given the classical argument of Mazur that T_{𝔪} is Gorenstein, but we still haven’t shown that J[𝔪] has dimension 2. This will be accomplished next time using Dieudonné modules.

18.4 Finite flat group schemes

Definition 18.4.1. Let S be a scheme. Then a group scheme over S is a group object in the category of S-schemes.

Thus a group scheme over S is a scheme G/S equipped with S-morphisms m : G × G → G, inv : G → G and a section 1_{G} : S → G satisfying the usual group axioms.

Suppose G is a group scheme over the finite field F_{q}. If R is an F_{q}-algebra then G(R) = Mor(Spec R, G) is a group. It is the group of R-valued points of G.

We consider several standard examples of group schemes. Example 18.4.2. The multiplicative group scheme G_{m} is G_{m} = Spec Z[x, \frac{1}{x}] with morphisms. [[give maps, etc.]] The additive group scheme is Spec Z[x]... Example 18.4.3. The group scheme μ_{p} is the kernel of the morphism G_{m} → G_{m} induced by x ↦ x^{p}. Thus μ_{p} = Spec Z[x]/(x^{p} − 1) and so for any F_{q}-algebra R we have that μ_{p}(R) = {r ∈ R : r^{p} = 1}. The group scheme α_{p} is the kernel of the morphism G_{a} → G_{a} induced by [[what!! what is alphap?? it should be the additive group scheme of order p, no?]].

Let A be a finite algebra over F_{p} and suppose G = Spec A affine group scheme (over F_{p}). Then the order of G is defined to be the dimension of A as an F_{p} vector space. Example 18.4.4. Let E/F_{p} be an elliptic curve. Then G = E[p] is a group scheme of order p^{2} [[why is this true?]]. This is wonderful because this is the order that E[p] should have in analogy with the characteristic 0 situation. When we just look at points we have

#G(\overline{F_{p}}) = \begin{cases}1 & supersingular \\ p & ordinary \end{cases} .

18.5 Reformulation of V = W problem

Let J = J_{0}(N) be the Jacobian of X_{0}(N). Then J is defined over Q and has good reduction at all primes not dividing N. Assume ℓ is a prime not dividing N. J[ℓ] extends to a finite flat group scheme over Z[\frac{1}{N}]. This is a nontrivial result of Grothendieck (SGA 7I, LNM 288). Since ℓ ∤ N, J[ℓ] gives rise to a group scheme over F_{ℓ}.

We have “forcefully” constructed a Galois representation ρ_{𝔪} : G → V of dimension 2 over T/𝔪. Our goal is to show that this is isomorphic to the naturally defined Galois representation W = J[𝔪]. So far we know that

0 ⊂ V ⊂ W ⊂ J[ℓ].

Our assumptions are that ℓ ∤ N, V is irreducible, and ℓ ≠ 2.

Let \underline{J} be J thought of as a scheme over Z_{ℓ}. Grothendieck showed that \underline{J} is the spectrum of a free finite Z_{ℓ}-module. Raynaud (1974) showed that if ℓ ≠ 2 then essentially everything about \underline{J}[ℓ] can be seen in terms of J[ℓ](\overline{Q}_{ℓ}). He goes on to construct group scheme \underline{V} and \underline{W} over Z_{ℓ} such that

\underline{V} ⊂ \underline{W} ⊂ \underline{J}[ℓ].

Our goal is to prove that the inclusion V ↪ W of Galois modules is an isomorphism. Raynaud showed noted that the category of finite flat group schemes over Z_{ℓ} is an abelian category so the cokernel \underline{Q} = \underline{W} /\underline{V} is defined. Furthermore, V = W if and only if \underline{Q} = 0. Since \underline{Q} is flat \underline{Q}_{F_{ℓ}} has the same dimension over F_{ℓ} as \underline{Q}_{Q_{ℓ}} has over Q_{ℓ}. Passing to characteristic ℓ yields an exact sequence

0 → \underline{V}_{F_{ℓ}} → \underline{W}_{F_{ℓ}} → \underline{Q}_{F_{ℓ}} → 0.

Thus V ↪ W is an isomorphism if and only if \underline{V}_{F_{ℓ}} ↪ \underline{W}_{F_{ℓ}} is an isomorphism. Since \underline{V}, \underline{W}, and \underline{Q} have an action of k = T/𝔪 that are k-vector space schemes. This leads us to Dieudonné theory.

18.6 Dieudonné theory

Let G/k be a finite k-vector space scheme where k is a finite field of order q. Suppose G has order q^{n} so G is locally the spectrum of a rank n algebra over k. The Dieudonné functor contravariantly associates to G an n dimensional k-vector space D(G). Let Frob : G → G be the morphism induced by the pth power map on the underlying rings and let Ver be the dual of Frob. Let φ = D(Frob) and ν = D(Ver), then it is a property of the functor D that φ ∘ ν = ν ∘ φ = 0. The functor D is a fully faithful functor. Example 18.6.1. Let k = F_{p}. If G is either μ_{p}, α_{p}, or Z/pZ then D(G) is a onedimensional vector space over k. In the case of α_{p}, φ = ν = 0. For μ_{p}, φ = 0 and ν ≠ 1 and for Z/pZ, φ ≠ 1 and ν = 0. [[The latter two could be reversed!]]

Let G^{∨} = Hom(G, μ_{p}) denote the Cartier dual of the scheme G. Then

D(G^{∨}) = Hom_{k}(D(G), k)

(φ and ν are switched.) Example 18.6.2. Let A/F_{ℓ} be an abelian variety and let G = A[ℓ]. Then G is an F_{ℓ}-vector space scheme of order ℓ^{2g}. Thus D(G) is a 2g-dimensional F_{ℓ}-vector space and furthermore D(G) = H_{DR}^{1}(A/F_{ℓ}). The Hodge filtration on H_{DR}^{1} of the abelian variety A gives rise to a diagram

[figure: Hom(H^{0}(A^{∨}, Ω^{1}), F_{ℓ}); Tan(A^{∨}); H^{0}(A, Ω^{1}); D(G); H_{DR}^{1}(A/F_{ℓ}); H^{1}(A, O); D(A[ℓ]); D(A[Ver])]

There is an exact sequence

0 → W_{F_{ℓ}} → J_{F_{ℓ}}[ℓ] ⟶^{𝔪} J_{F_{ℓ}}[ℓ]

so because D is an exact functor the sequence

D(J_{F_{ℓ}}[ℓ]) ⟶^{𝔪} D(J_{F_{ℓ}}[ℓ]) → D(W_{F_{ℓ}}) → 0

is exact. Following Fontaine we consider

D(W_{F_{ℓ}}[Ver]) = H^{1}(J, O)/𝔪H^{1}(J, O).

18.7 The proof: part II

[[We all just returned from the Washington D.C. conference and will now resume the proof.]]

Let J_{0}(N) be the Jacobian of X_{0}(N). Let 𝔪 ⊂ T be a maximal ideal and suppose 𝔪|ℓ. Assume that ℓ ≠ 2 and ℓ ∤ N. The assumption that ℓ ≠ 2 is necessary for Raynaud’s theory and we assume that ℓ ∤ N so that our group schemes will have good reduction. We attach to 𝔪 a 2 dimensional semisimple representation ρ_{𝔪} : Gal(\overline{Q}/Q) → V and only consider the case that ρ_{𝔪} is irreducible.

The 𝔪-torsion of the Jacobian, W = J_{0}(\overline{Q})[𝔪], is naturally a Galois module. We have shown that W ≠ 0. By [BpR91] W ≅ V × ⋯ × V (the number of fractions is not determined). We proved that W^{s.s.} ≅ V × ⋯ × V. Choose an inclusion V ↪ W and let Q = W/V be the cokernel.

Theorem 18.7.1. Q = 0 so dim_{T/𝔪} W = 2

To prove the theorem we introduce the “machine” of finite flat group schemes over Z_{ℓ}. For example, W extends to a finite flat group scheme W_{Z_{ℓ}} which is defined to be the Zariski closure of W in J_{Z_{ℓ}}[ℓ]. Passing to group schemes yields an exact sequence

0 → V_{Z_{ℓ}} → W_{Z_{ℓ}} → Q_{Z_{ℓ}} → 0.

Reducing mod ℓ then yields an exact sequence of F_{ℓ}-group schemes

0 → V_{F_{ℓ}} → W_{F_{ℓ}} → Q_{F_{ℓ}} → 0.

The point is that Q = 0 if and only if Q_{Z_{ℓ}} = 0 if and only if Q_{F_{ℓ}} = 0. Next we introduced the exact contravariant Dieudonné functor

D : ( Groups Schemes /F_{ℓ} ) ⟶ ( Linear Algebra ).

D sends a group scheme G to a T/𝔪 vector space equipped with two endomorphisms φ = Frob and ν = Ver. Applying D gives an exact sequence of T/𝔪-vector spaces

0 → D(Q) → D(W) → D(V) → 0

where everything is now viewed over F_{ℓ}.

Lemma 18.7.2. D(W [Ver]) = (H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[𝔪])^{∗}

Proof. We have the diagram

D(J_{F_{ℓ}}[ℓ]) =  H_{DR}^{1}(J_{F_{ℓ}})
↓  ↓
D(J_{F_{ℓ}}[Ver]) =  H^{1}(J_{F_{ℓ}}, O_{J_{F_{ℓ}}})
↓  ↓
D(W [Ver]) = H^{1}(J_{F_{ℓ}}, O_{J})/𝔪H^{1}(J_{F_{ℓ}}, O_{J})

Furthermore we have the identifications

H^{1}(J_{F_{ℓ}}, O_{J}) = Tan(J_{F_{ℓ}}^{∨}) = Cot(J_{F_{ℓ}}^{∨})^{∗}
= H^{0}(J^{∨}, Ω^{1})^{∗} = H^{0}(X_{0}(N), Ω^{1})^{∗}

For the last identification we must have J^{∨} = Alb(X_{0}(N)). Finally

D(W [Ver]) = H^{1}(J, O_{J})/𝔪H^{1}(J, O_{J})
= (H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[𝔪])^{∗}.

Lemma 18.7.3. H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[𝔪] has T/𝔪 dimension ≤ 1.

Proof. Let S = H^{0}(X_{0}(N)_{F_{ℓ}}, Ω^{1})[𝔪]. Then S ↪ F_{ℓ}[[q]]. We defined the Hecke operators T_{n} on S via the identification S ≅ H^{1}(J, O_{J}) so that they act on S ⊂ F_{ℓ}[[q]] in the standard way. Let T(S) be the subalgebra of End(S) generated by the images of the T_{n} in End(S). (T(S) is not a subring of T.) There is a perfect pairing

T(S) × S ⟶ F_{ℓ}
(T, f) ↦ a_{1}(f|T)

Thus dim_{F_{ℓ}} T(S) = dim_{F_{ℓ}} S and so dim_{T(S)} S ≤ 1. Since 𝔪 acts trivially on S there is a surjection T/𝔪 → → T(S). Thus

dim_{T/𝔪} S ≤ dim_{T(S)} S ≤ 1

as desired.

An application of the above lemma shows that D(W [Ver]) has T/𝔪 dimension ≤ 1.

Lemma 18.7.4. D(W [Ver]) ≅ D(V [Ver])

Proof. Consider the following diagram.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & D(Q) \ar[r] \ar[d]^{Ver} & D(W) \ar[r] \ar[d]^{Ver} & D(V) \ar[r] \ar[d]^{Ver} & 0 \\ 0 \ar[r] & D(Q) \ar[r] \ar[d] & D(W) \ar[r] \ar[d] & D(V) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & D(Q)/ Ver D(Q) \ar[d] \ar[r] & D(W)/ Ver D(W) \ar[d] \ar[r]^{?≅?} & D(V)/ Ver D(V) \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

D(V) has dimension 2 so since Ver ∘ Frob = Frob ∘ Ver = 0 and Ver, Frob are both nonzero they must each have rank 1 (in the sense of undergraduate linear algebra). Since D is exact, D(V [Ver]) = D(V)/ Ver D(V) and D(W [Ver]) = D(W)/ Ver D(W). By the previous lemma dim D(W)/ Ver D(W) = 1. Thus D(W)/ Ver D(W) → D(V)/ Ver D(V) is a map of 1 dimensional vector spaces so to show that it is an isomorphism we just need to show that it is surjective. This follows from the commutativity of the square

\xymatrix{D(W) \ar[r] \ar[d] & D(V) \ar[r] \ar[d] & 0 \\ D(W)/ Ver D(W) \ar[r] & D(V)/ Ver D(V) \ar[d] \\  & 0}

Suppose for the moment that we admit [BpR91]. Then

W = V × ⋯ × V = V^{⊕t}

so

D(W [Ver]) ≅ D(V [Ver])^{⊕t}

and hence t = 1.

Alternatively we can avoid the use of [BpR91]. Suppose Q ≠ 0. Then there is an injection V ↪ Q. [[I can’t see this without using B-L-R. It isn’t obvious to me from 0 → V → W → Q → 0.]] Thus over F_{ℓ}, V [Ver] ↪ Q[Ver]. Since V [Ver] ≠ 0 this implies Q[Ver] ≠ 0. Thus D(Q)/ Ver D(Q) = D(Q[Ver]) ≠ 0. But the bottom row of the above diagram implies D(Q)/ Ver D(Q) = 0 so Q = 0.

18.8 Key result of Boston-Lenstra-Ribet

Let G be a group (i.e., G = Gal(\overline{Q}/Q)), let k be a field (i.e., k = T/𝔪), and let V be a two dimensional k-representation of G given by

ρ : k[G] → End_{k}(V) = M_{2}(k).

The key hypothesis is that V is absolutely irreducible, i.e., that ρ is surjective. For each g ∈ G consider

p_{g} = g^{2} − g tr ρ(g) + det ρ(g) ∈ k[G].

By the Cayley-Hamilton theorem ρ(p_{g}) = 0. Let J be the two-sided ideal of k[G] generated by all p_{g} such that g ∈ G. Since J ⊂ ker ρ , ρ induces a map

σ : k[G]/J → End_{k}(V).

Theorem 18.8.1 (Boston-Lenstra-Ribet). If σ is surjective then σ is an isomorphism.

In particular if V is absolutely irreducible then σ is surjective. The theorem can be false when dim V > 2.

Suppose W is a second representation of G given by μ : k[G] → End(W) and that μ(J) = {0} ⊂ End(W). Then W is a module over k[G]/J = End(V). But End(V) is a semisimple ring so any End(V) module is a direct sum of simple End(V) modules. The only simple End(V) module is V. Thus W ≅ V^{⊕^{n}} for some n.

194 18. The Gorenstein Property for Hecke Algebras

Local Properties of ρ_{λ}

Let f be a newform of weight 2 on Γ_{0}(N). To f we have associated an abelian variety A = A_{f} furnished with an action of E = Q(…, a_{n}(f), …). Let O_{E} be the ring of integers of E and λ ⊂ O_{E} a prime. Then we obtain a λ-adic representation

ρ_{λ} : Gal(\overline{Q}/Q) → GL_{2}(E_{λ})

on the Tate module Tate_{ℓ} A = \underleftarrow{lim} A[λ^{i}]. We will study the local properties of ρ_{λ} at various primes p.

19.1 Definitions

To view ρ_{λ} locally at p we restrict to the decomposition group D_{p} = Gal(\overline{Q}/Q) at p. Recall the definition of D_{p}. Let K be a finite extension of Q and let w be a prime of K lying over p. Then the decomposition group at w is defined to be

D_{w} = {σ ∈ Gal(K/Q) : σw = w}.

Proposition 19.1.1. D_{w} ≅ Gal(K_{w}/Q_{p})

Proof. Define a map Gal(K_{w}/Q_{p}) → D_{w} by σ ↦ σ_{|K}. Since σ_{|K} fixes Q this restriction is an element of Gal(K/Q). Since wO_{K_{w}} is the unique maximal ideal of O_{K_{w}} and σ induces an automorphism of O_{K_{w}}, it follows that σ(wO_{K_{w}}) = wO_{K_{w}}. Thus σ_{|K}(w) = w so σ_{|K} ∈ D_{w}. The map σ ↦ σ_{|K} is bijective because K is dense in K_{w}.

Let

D_{p} = \underleftarrow{lim}_{w|p} D_{w} = \underleftarrow{lim}_{w|p} Gal(K_{w}/Q_{p}) = Gal(\overline{Q}_{p}/Q_{p}).

For each w|p let the inertia group I_{w} be the kernel of the map from D_{w} into Gal(O_{K}/w, Z/pZ). Let I_{p} = \underleftarrow{lim}_{w|p} I_{w}.

196 19. Local Properties of ρ_{λ}

19.2 Local properties at primes p ∤ N

Next we study local properties of ρ_{λ} at primes p ∤ N. Thus suppose p ∤ N and p ≠ ℓ = char(O_{E}/λ). Let D_{p} = Gal(\overline{Q}_{p}/Q_{p}). Then

1) ρ_{λ}|D_{p} is unramified (i.e., ρ_{λ}(I_{p}) = {1}) thus ρ_{λ}|D_{p} factors through D_{p}/I_{p} so ρ_{λ}(Frob_{p}) is defined.

2) tr(ρ_{λ}(Frob_{p})) = a_{p}(f)

2+) We can describe ρ_{λ}|D_{p} up to isomorphism. It is the unique semisimple representation satisfying 1) and 2).

19.3 Weil-Deligne Groups

Notice that everything is sort of independent of λ. Using Weil-Deligne groups we can summarize all of these λ-adic representations in terms of data which makes λ disappear.

We have an exact sequence

1 → I_{p} → Gal(\overline{Q}_{p}/Q_{p}) → Gal(\overline{F}_{p}/F_{p}) → 0.

Since Gal(\overline{F}_{p}/F_{p}) = Ẑ there is an injection Z ↪ Gal(\overline{F}_{p}/F_{p}). Define the Weil group W (\overline{Q}_{p}/Q_{p}) ⊂ Gal(\overline{Q}_{p}/Q_{p}) to be the set of elements of Gal(\overline{Q}_{p}/Q_{p}) mapping to Z ⊂ Gal(\overline{F}_{p}/F_{p}). W fits into the exact sequence

1 → I_{p} → W (\overline{Q}_{p}/Q_{p}) → Z → 1.

There is a standard way in which the newform f gives rise to a representation of W. Factor the polynomial x^{2} − a_{p}(f)x + p as a product (x − r)(x − r′) with r, r′ ∈ C. Define maps α, β by

α : Z → C^{∗} : 1 ↦ r
β : Z → C^{∗} : 1 ↦ r′

Combining α and β and the map W (\overline{Q}_{p}/Q_{p}) → Z yields a map

α ⊕ β : W (\overline{Q}_{p}/Q_{p}) → GL_{2}(C)

σ ↦ (\begin{matrix}α(σ) & 0 \\ 0 & β(σ) \end{matrix}).

Moreover α ⊕ β gives rise via some construction to all the λ-adic representations ρ_{λ}.

19.4 Local properties at primes p | N

Suppose p|N but p ≠ ℓ. Carayol was able to generalize 1) in his thesis which builds upon the work of Langlands and Deligne in the direction of Deligne-Rapaport and Katz-Mazur. The idea is that the abelian variety A has a conductor which is a positive integer divisible by those primes of bad reduction. The conductor of A satisfies cond(A) = M^{g} where g = dim A and M is the reduced conductor of A.

Theorem 19.4.1 (Carayol). M = N.

How can we generalize 2) or 2+)? For each p dividing N there is a representation σ_{p} of WD(\overline{Q}_{p}/Q_{p}) over C such that σ_{p} gives rise to ρ_{λ}|D_{p} for all λ ∤ p. Here WD(\overline{Q}_{p}/Q_{p}) is the Weil-Deligne group which is Deligne’s generalization of the Weil group. WD(\overline{Q}_{p}/Q_{p}) is an extension of W (\overline{Q}_{p}/Q_{p}). What is σ_{p} supposed to be? The point is that σ_{p} is determined by f. Thus every f gives rise to a family (σ_{p})_{p prime}. To really think about σ_{p} we must think about modular forms in an adelic context instead of viewing them as holomorphic functions on the complex upper halfplane.

If p^{2}|N and f = ∑ a_{n}q^{n} is a newform of level p^{2} then it is a classical fact that a_{p} = 0. But the study of ρ_{λ}|D_{p} is rich and “corresponds to a rather innocuous looking crystal”.

19.5 Definition of the reduced conductor

We now define the reduced conductor. Let λ be a prime of E and p a prime of Q such that λ ∤ p. We want to define some integer e(p) so that p^{e(p)} is the p-part of the reduced conductor. We will not define e(p) but what we will do is define an integer e(p, λ) which is the p part of the conductor. e(p, λ) is independent of λ but this will not be proved here.

Consider

ρ_{λ}|D_{p} : Gal(\overline{Q}_{p}/Q_{p}) → Aut_{E_{λ}} V.

Let V^{I} ⊂ V be the inertia invariants of V, i.e.,

V^{I} = {v ∈ V : ρ_{λ}(σ)(v) = v for all σ ∈ I}.

Since ρ_{λ} is unramified at p if and only if ρ_{λ}(I_{p}) = {1} we comment that ρ_{λ} is unramified at p if and only if V^{I} = V. Let

e(p, λ) = dim V/V^{I} + δ(p, λ)

where δ(p, λ) is the Swan conductor. By working with finite representations we define δ(p, λ) as follows. Choose a D_{p}-stable lattice L in V by first choosing an arbitrary one then taking the sum of its finitely many conjugates. Let \overline{V} = L/λL which is a 2 dimensional vector space over k = O_{E}/λ. Let G be the quotient of Gal(\overline{Q}_{p}/Q_{p}) by the kernel of the map Gal(\overline{Q}_{p}/Q_{p}) → Aut_{K} \overline{V}. Thus G = Gal(K/Q_{p}) for some finite extension K/Q_{p}. The extension K is finite over Q_{p} since G ⊂ Aut_{k} \overline{V} and Aut_{k} \overline{V} is a 2 × 2 matrix ring over a finite field. The corresponding diagram is

\xymatrix{Gal(\overline{Q}_{p}/Q_{p}) \ar[r] \ar@{-->}[dr] & Aut_{k} \overline{V} \\  & G = Gal(K/Q_{p}) \ar[u]}

Consider in G the sequence of “higher ramification groups”

G = G_{−1} ⊃ G_{0} ⊃ G_{1} ⊃ · · · .

Here G_{0} is the inertia group of K/Q_{p} and G_{1} is the p-sylow subgroup of G_{0} [[the usages of “the” in this sentence makes me nervous.]] Let G_{i} = {g ∈ G_{0} : ord(gπ − π) ≥ i + 1} where π is some kind of uniformizing parameter [[I missed this – what is π?]] Let

δ(p, λ) = ∑_{i=1}^{∞} \frac{1}{(G_{0} : G_{i})} dim(\overline{V} /\overline{V}^{G_{i}}).

It is a theorem that δ(p, λ) is an integer and does not depend on λ.

If to start with we only had \overline{V} and not V we could define

cond(\overline{V}) = ∑_{i=0}^{∞} \frac{1}{(G_{0} : G_{i})} dim(\overline{V} /\overline{V}^{G_{i}}).

Then

e(p, λ) = cond(\overline{V}) + (dim \overline{V}^{I} − dim V^{I}).

A reference for much of this material is Serre’s Local Factors of L-functions of λ-adic Representations.

Adelic Representations

Our goal is to study local properties of the λ-adic representations ρ_{λ} arising from a weight 2 newform of level N on Γ_{0}(N). There is a theorem of Carayol which states roughly that if p ≠ ℓ then ρ_{λ}|D_{p} is predictable from “the component at p of f”. To understand this theorem we must understand what is meant by “the component at p of f”. If p^{2} ∤ N this component is easy to determine but if p^{2}|N it is harder. One reason is that when p^{2}|N then a_{p}(f) = 0. [[this should be easy to see so there should be an argument here.]] If a_{p}(f) = 0 and ρ_{λ} : Gal(\overline{Q}/Q) → Aut(V_{λ}) then V_{λ}^{I_{p}} = 0. This means that there is no ramification going on at p. See Casselman, “On representations of GL(2) and the arithmetic of modular curves”, Antwerp II.

20.1 Adelic representations associated to modular forms

Let R be a subring of A^{2} = R^{2} × (Ẑ ⊗_{Z} Q)^{2}, suppose that R ≅ Q^{2} and that R ⊗_{Q} A ≅ A^{2}. Let L = R ∩ (R^{2} × Ẑ^{2}). Then the natural map L ⊗ Ẑ → Z^{2} is an isomorphism. [[Is the isomorphism implied by the definition of L or is it part of the requirement for L to actually form an adelic lattice?]] L is called an adelic lattice.

The space of modular forms S_{2}(Γ_{0}(N)) is isomorphic to a certain space of functions on G(A) = GL(2, A). See Borel-Jacquet [[Corvallis?]] or Diamond-Taylor, Inventiones Mathematica, 115 (1994) [[what is title?]] We will describe this isomorphism.

Write A = R × A_{f} = A_{∞} × A^{∞} where A_{f} = ∏ Q_{p} (restricted product) is the ring of finite adeles. A_{∞} denotes the adeles with respect to the place ∞ so A_{∞} = R, and A^{∞} denotes the adeles away from the place ∞ so A^{∞} = A_{f}.

S_{2}(Γ_{0}(N)) is isomorphic to the set of functions φ : G(A) → C which satisfy

0) φ is left invariant by G(Q), i.e., φ(x) = φ(gx) for all g ∈ G(Q) and all x ∈ G(A), 1) φ(xu^{∞}) = φ(x) for all x ∈ G(A) and all u^{∞} ∈ U^{∞},

2) φ(xu_{∞}) = φ(x)j(u_{∞}, i)^{−k} det(u_{∞}) for all x ∈ G(A) and u_{∞} ∈ U_{∞},

3) holomorphy, cuspidal, and growth conditions.

U^{∞} is the adelic version of Γ_{0}(N). Thus U^{∞} is the compact open subgroup

U^{∞} = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(Ẑ) : c ≡ 0 (mod N)\} ⊂ G(A_{f}).

(For Γ_{1}(N) the condition is that c ≡ 0 (mod N) and d ≡ 1 (mod N) but a is not restricted.)

Next we describe U_{∞} ⊂ GL_{2}(R). GL_{2}(R) operates on 𝔥^{±} = C−R by z ↦ \frac{az+b}{cz+d}. Let U_{∞} be the stabilizer of i.

The third condition involves the automorphy factor j defined by

j((\begin{matrix}a & b \\ c & d \end{matrix}), z) = cz + d.

To explain the holomorphy condition 3) we define, for any g ∈ G(A_{f}) a map

α_{g,φ} : 𝔥^{±} → C
hi ↦ φ(gh)j(h, i)^{k}(det(h))^{−1}.

Here h ∈ G(A_{f}) so hi ∈ 𝔥^{±}. There may be several different h ∈ G(A_{f}) which give the same hi ∈ 𝔥^{±} so it must be checked that α_{g,φ} is well-defined. Suppose hi = h′i, then h^{−1}h′i = i. Thus h^{−1}h′ ∈ U_{∞}, so by 2),

φ(gh^{−1}h′) = φ(g)j(h^{−1}h′, i)^{−k} det(h^{−1}h′).

Thus

φ(gh^{−1}h′) det(h′)^{−1} = φ(g)j(h^{−1}h′, i)^{−k} det(h)^{−1}.

Substituting gh for g yields

φ(gh′)(det(h′))^{−1} = φ(gh)j(h^{−1}h′, i)^{−k}(det(h))^{−1}.

[[The the automorphy factor work out right. Why?]] The holomorphy condition is that the family of maps α_{g,φ} are all holomorphic.

The cuspidal condition is that for all g ∈ G(A), the integral

∫_{u∈A/Q} φ(( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ), g)du = 0

vanishes. Since A/Q is compact it has a Haar measure defined modulo k^{∗} which induces du. Although the integral is not well-defined the vanishing or non-vanishing of the integral is.

We can now describe the isomorphism between S_{k}(Γ_{0}(N)) and the space of such functions φ on G(A).

{ space of φ satisfying 0-3} → S_{k}(Γ_{0}(N))
φ ↦ f

where f is the restriction to 𝔥 = 𝔥^{+} of the function

hi ↦ φ(h)j(h, i)^{k}(det h)^{−1}.

Now we can associate to a newform f a representation of G(A). We can weaken condition 1) to get

1-) φ(xu^{∞}) = φ(x) for all x ∈ G(A) and all u^{∞} ∈ U^{∞} where U^{∞} is some compact open subgroup of G(A_{f}).

[[can the U^{∞} vary for each x ∈ G(A) or are they fixed throughout?]] Let S be the space of all functions satisfying all conditions except 1-) replaces 1). This space has a left action of G(A):

(g ∗ φ)(x) = φ(xg)

If f is a newform corresponding to some φ via the above isomorphism then via this action f gives rise to an infinite dimensional representation π of G(A). In fact we obtain, for each prime p, a representation π_{p} of GL(2, Q_{p}). The representation space is ∑_{g∈GL_{2}(Q_{p})} C · g ∗ φ. Our immediate goal is to understand π_{p} for as many p as possible. [[spherical representations have something to do with this. are the π_{p} spherical reps?]]

We are studying local properties of the λ-adic representations ρ_{λ} associated to a newform f of weight 2, level N and character ε : (Z/NZ)^{∗} → C^{∗} (with cond(ε)|N). Let ℓ ∈ Z be the prime over which λ lies. We look at ρ_{λ} locally at p, p ≠ ℓ.

As we saw last time f gives rise to an irreducible representation of GL(2, A). An irreducible representation of GL(2, A) gives rise to a family of representations (π_{v}) where v is a prime or ∞ and π_{v} is an irreducible representation of GL(2, Q_{v}). This is because Q_{v} ⊂ A so GL(2, Q_{v}) ⊂ GL(2, A).

Carayol proved that if p ≠ ℓ then ρ_{λ}|_{D_{p}} depends, up to isomorphism, only on π_{p}. The most difficult case in the proof of this theorem is when p^{2} ∤ N. The tools needed to obtain a proof were already available in the work of Langlands [1972].

To get an idea of what is going on we will first consider the case when p ∤ N. The characteristic polynomial of Frobenious (at least psychologically) under the representation ρ_{λ}|_{D_{p}} is

x^{2} − a_{p}x + pε(p) = (x − r)(x − s).

Because of Weil’s proof of the Riemann hypothesis for abelian varieties [over finite fields?] one knows that |r| = |s| = \sqrt{p}. Since ρ_{λ} arises from the action of Galois on an abelian variety which has good reduction at p (since p ∤ N) it follows that ρ_{λ}|_{D_{p}} is unramified. [[Is this in Serre-Tate, 1968?]] We also know that ρ_{λ}(Frob_{p}) has characteristic polynomial

x^{2} − a_{p}x + pε(p) ∈ E[x].

In this situation one also knows that ρ_{λ}(Frob_{p}) is semisimple [[proof: Ribet nodded at Coleman who smiled at nodded back.]] Thus

ρ_{λ} ∼ (\begin{matrix}r & 0 \\ 0 & s \end{matrix}).

In this situation what is the representation π_{p} of GL(2, Q_{p})? There are two characters

α, β : Q_{p}^{∗} → C^{∗}

(called “Grössencharacters of type (a,0)”) such that 1. α and β are unramified in the sense that

α|_{Z_{p}^{∗}} = β|_{Z_{p}^{∗}} = 1.

This is a reasonable condition since under some sort of local class field theory Q_{p}^{∗} embeds as a dense subgroup of Gal(\overline{Q}_{p}/Q_{p}) and under this embedding the inertia subgroup I ⊂ Gal(\overline{Q}_{p}/Q_{p}) corresponds to Z_{p}^{∗}. [[This could be wrong. Also, is I ∩ Q_{p}^{∗} = Z_{p}^{∗}?]]

2. α(p^{−1}) = r, β(p^{−1}) = s.

In the 1950’s Weil found a way under which α and β correspond to continuous characters α_{λ}, β_{λ} on Gal(\overline{Q}_{p}/Q_{p}) with values in \overline{E}_{λ}^{∗} such that

1. α_{λ} and β_{λ} are unramified.

2. α_{λ}(Frob_{p}) = r and β_{λ}(Frob_{p}) = s.

One has that ρ_{λ} = α_{λ} ⊕ β_{λ}. See [ST68]. [[Why see this?]] Define a character Θ on the Borel subgroup

B = (\begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix}) ⊂ GL(2, Q_{p})

by

Θ(\begin{matrix}x & y \\ 0 & z \end{matrix}) = α(x)β(z) ∈ C^{∗}.

Then

π_{p} = Ind_{B}^{GL(2,Q_{p})} := C[GL(2, Q_{p})] ⊗_{C[B]} C.

We call this induced representation π_{p} the unramified principal series representation associated to α and β and write π_{p} = PS(α, β). People say π_{p} is spherical in the sense that there is a vector in the representation space invariant under the maximal compact subgroup of GL(2, Q_{p}). [[Ribet was slightly unsure about the correct definition of spherical.]] [[For some mysterious reason]] since π_{p} = PS(α, β) it follows that α, β and hence ρ_{λ}|D_{p} is completely determined by π_{p}. [[This is the point and i don’t see this.]]

Next we consider the more difficult case when p||N (p divides N exactly). There are two cases to consider

a ε is ramified at p (p| cond(ε))

b ε is unramified at p (p ∤ cond(ε)).

20.2 More local properties of the ρ_{λ}.

20.2 More local properties of the ρ_{λ}. 203

Let f be a newform of level N. Then f corresponds to a representation π = ⊗π_{v}. If λ|ℓ and ℓ ≠ p then ρ_{λ}|D_{p} corresponds to π_{p} under the Langlands correspondence. The details of this correspondence were figured out by Philip Kutzko, but Carayol completed it in the exceptional case (to be defined later). There are three cases to consider. a) p ∤ N

b) p||N

c) p^{2} | N

We considered the first two cases last time. The third case is different because the same sort of analysis as we applied to the first two cases no longer works in the sense that we no longer know what π_{p} looks like.

20.2.1 Possibilities for π_{p}

Case 1 (principal series) In this case, π_{p} = PS(α, β). Here α, β : Q_{p}^{∗} → C^{∗} are unramified characters. Q_{p}^{∗} corresponds to a dense subgroup of the abelian Galois group of Q_{p} under the correspondence elucidated in Serre’s “Local Classfield Theory” (in Cassels and Frohlich).

\xymatrix{Q_{p}^{∗} \ar[r] \ar[d] & Gal(\overline{Q}_{p}/Q_{p})^{A∩B} \ar[d] \\ Z \ar[r] & Ẑ = Gal(\overline{F}_{p}/F_{p})}

Under this correspondence α and β correspond to Galois representations α_{λ} and β_{λ} and ρ_{λ}|D_{p} ∼ α_{λ} ⊕ β_{λ}.

Case 2 (special) In this case, π_{p} is the special automorphic representation corresponding to the Galois representation κ ⊗ st where st is the Steinberg representation which arises somehow from a split-multiplicative reduction elliptic curve and κ is a Dirichlet character. In this case

ρ_{λ}|D_{p} = κ ⊗ (\begin{matrix}χ_{ℓ} & ∗ \\ 0 & 1 \end{matrix}).

Case 3 (cuspidal) Case 3 occurs when π_{p} does not fall into either of the previous cases. Such a π_{p} is called cuspidal or super-cuspidal. Some of these π_{p} come from the following recipe. Fix an algebraic closure \overline{Q}_{p} of Q_{p}. Let K be a quadratic extension of Q_{p}. Let ψ : K^{∗} → C^{∗} be a Grössencharacter. Then ψ gives rise to a character

ψ_{λ} : Gal(\overline{Q}_{p}/K) → \overline{E}_{λ}^{∗}

which induces ρ_{λ}|D_{p}. That is,

ρ_{λ}|D_{p} = Ind_{Gal(\overline{Q}_{p}/K)}^{Gal(\overline{Q}_{p}/Q_{p})} ψ_{λ} : Gal(\overline{Q}_{p}/Q_{p}) → GL(2, \overline{E}_{λ}).

This representation is irreducible if and only if ψ is not invariant under the canonical conjugation of K/Q_{p}. The pair ψ, K gives rise (via the construction of Jacquet-Langlands) to a representation of π_{ψ,K} of GL(2, Q_{p}). Those representations which do not come from this recipe and which do not fall into case 1 or case 2 above are called extraordinary. They can only occur when p ≤ 2.

When can the various cases occur?

Case 1) occurs, e.g, if p ∤ N, and also if p||N and ε (the character of f) is ramified at p.

Case 2) occurs if p||N and ε is unramified at p.

20.2.2 The case ℓ = p

We consider ρ_{λ}|D_{p} where λ|p and p = ℓ. Write f = ∑ a_{n}q^{n}. The case when λ ∤ a_{p} is called the ordinary case. This case is very similar to the case for an ordinary elliptic curve. In other words,

ρ_{λ}|D_{p} = (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

[[Ribet mentioned that there is a paper on this by Mazur and Wiles in the American Journal of Mathematics. Look the reference up.]] An important point is that β is unramified so it makes sense to consider β(Frob_{p}) = a_{p} ∈ E_{λ}^{∗}. Since αβ is the determinant, αβ = χ_{p}^{k−1}ε = χ_{p}ε (after setting k = 2 to fix ideas), this gives some description of what is going on. The obvious question to ask is whether or not * is nontrivial. That is, is ρ_{λ}|D_{p} semisimple or not?

When f has weight 2, then f gives rise to an abelian variety A = A_{f}. Then ρ_{λ} is defined by looking at the action of Galois on the λ-adic division points on A. If none of the λ lying over p divide a_{p} then A is ordinary at p. A stronger statement is that the p-divisible group A[p^{∞}] has good ordinary reduction.

One simple case is when f has CM. By this we mean that there is a character κ ≠ 1 of order 2 such that a_{n} = κ(n)a_{n} for all n prime to cond(κ). [[This is not a typo, I do not mean \overline{a_{n}} = κ(n)a_{n}. Coleman said that a_{n} = κ(n)a_{n} is just a funny way to say that half of the a_{n} are 0.]] It is easy to prove that f has CM if and only if the ρ_{λ} become abelian on some open subgroup of Gal(\overline{Q}/Q) of finite index. Ribet explains this in his article in [Rib77]. If f has CM then since the representation ρ_{λ} is almost abelian one can show that * is trivial. Ribet said he does not know whether the converse is true. Note that f has CM if and only if A_{f}/\overline{Q} has CM. If all λ|p are ordinary (i.e., they do not divide a_{p}) and if * is trivial for every ρ_{λ}|D_{p} it is easy to show using [Ser98] that A has CM.

Next we will say something more about representations which appear to be ordinary. Consider the situation in which f has weight 2 and p exactly divides the level N of f. Suppose furthermore that the character ε of f is unramified at p. Then π_{p} is a special representation. The λ-adic representations for ℓ ≠ p are (up to characters of finite order) like representations attached to some Tate curve. The situation is similar when ℓ = p since

ρ_{λ}|D_{p} = (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

As in the case of a Tate curve α/β = χ_{p}. Up to a quadratic character we know the situation since αβ = χ_{p}ε. Also β is still unramified and β(Frob_{p}) = a_{p} is a unit. We know that a_{p}^{2} = ε(p) is a root of unity. When k = 2 the case of a spherical representation mimics what happens for ordinary reduction. The upper right hand entry * is never trivial in this case [[because of something to do with extensions and Kummer theory]].

Ribet said he knows nothing about the situation when k > 2. If p||N and ε is unramified at p then π_{p} is special so ρ_{λ}, ℓ ≠ p are again of the form ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix} ). So we still know everything up to a quadratic character. But if ℓ = p some characters are Hodge-Tate [[what does that mean?]] so by a theorem of Faltings they can not be bizarre powers of the cyclotomic character. The multiplicative case like the ordinary case is very special to the case k = 2. Wiles uses this heavily in his proof.

If k is arbitrary then a_{p}^{2} = ε(p)p^{\frac{k}{2} −1} so a_{p} is not a unit for k > 2 so there is no invariant line. So the representation is probably irreducible in the case k > 2. [[Echos of “yeah”, “strange” and “very strange” are heard throughout the room.]]

20.2.3 Tate curves

Suppose E/Q_{p} is an elliptic curve with multiplicative reduction at p and that j ∈ Q_{p} is the j-invariant of E. Using a formula which can be found in [Sil94, V] one finds q = q(j) with |q| < 1. The Tate curve is E(\overline{Q}_{p}) = \overline{Q}_{p}/q^{Z}. The p torsion on the Tate curve is {ζ_{p}^{a}(q^{1/p})^{b} : 0 ≤ a, b ≤ p − 1}. Galois acts by ζ_{p} ↦ ζ_{p}^{a} and q^{1/p} ↦ ζ_{p}^{a}q^{1/p}. Thus the associated Galois representation is ( \begin{matrix}χ_{p} & ∗ \\ 0 & 1 \end{matrix} ).

206 20. Adelic Representations

Serre’s Conjecture

This is version 0.2 of this section. I am still unsatisfied with the organization, level of detail, and coherence of the presentation. A lot of work remains to be done. Some of Ken’s lectures in Utah will be integrated into this chapter as well.

Let ℓ be a prime number. In this chapter we study certain mod ℓ Galois representations, by which we mean continuous homomorphisms ρ

\xymatrix{Gal(\overline{Q}/Q) \ar[r]^{ρ} & GL(2, F)}

where F is a finite field of characteristic ℓ. Modular forms give rise to a large class of such representations.

[figure: newforms f; mod ℓ representations ρ]

The motiving question is:

Given a mod ℓ Galois representation ρ, which newforms f if any, of various weight and level, give rise to ρ?

We will assume that ρ is irreducible. It is nevertheless sometimes fruitful to consider the reducible case (see [SW97] and forthcoming work of C. Skinner and A. Wiles).

Serre [Ser87] has given a very precise conjectural answer to our motivating question. The result, after much work by many mathematicians, is that certain of Serre’s conjectures are valid in the sense that if ρ arises from a modular form at all, then it arises from one having a level and weight as predicted by Serre. The main trends in the subject are “raising” and “lowering.”

Our motivating question can also be viewed through the opposite optic. What is the most bizarre kind of modular form that gives rise to ρ? A close study of the ramification behavior of ρ allows one to at least obtain some sort of control over the possible weights and levels. This viewpoint appears in [Wil95].

In this chapter whenever we speak of a Galois representation the homomorphism is assumed to be continuous. The reader is assumed to be familiar with local fields, some representation theory, and some facts about newforms and their characters.

21.1 The Family of λ-adic representations attached to a newform

First we briefly review the representations attached to a given newform. Let

f = ∑_{n≥1} a_{n}q^{n} ∈ S_{k}(N, ε)

be a newform of level N, weight k, and character ε. Set K = Q(…, a_{n}, …) and let O be the ring of integers of K. If λ is a nonzero prime ideal of O we always let ℓ be the prime of Z over which λ lies, so λ ∩ Z = (ℓ). Let K_{λ} denote the completion of K at λ, thus K_{λ} is a finite extension of Q_{ℓ}. The newform f gives rise to a system (ρ_{f,λ}) of λ-adic representations

ρ_{f,λ} : Gal(\overline{Q}/Q) ⟶ GL(2, K_{λ})

one for each λ.

Theorem 21.1.1 (Carayol, Deligne, Serre). Let f be as above and ℓ a prime. There exists a Galois representation

ρ_{f,ℓ} : Gal(\overline{Q}/Q) ⟶ GL(2, K ⊗ Q_{ℓ})

with the following property: If p ∤ ℓN is a prime, then ρ_{f,ℓ} is unramified at p, and the image under ρ_{f,ℓ} of any Frobenius element for p is a matrix with trace a_{p} and determinant ε(p)p^{k−1}.

The actual construction of ρ_{f,ℓ} won’t be used in what follows. Since K ⊗ Q_{ℓ} is a product ∏_{λ} K_{λ} of the various completions of K at the primes λ of K lying over ℓ, we have a decomposition

GL(2, K ⊗ Q_{ℓ}) = ∏_{λ|ℓ} GL(2, K_{λ})

and projection onto GL(2, K_{λ}) gives ρ_{f,λ}.

By Lemma 18.1.1 ρ_{f,λ} is equivalent to a representation taking values in GL(2, O) where O is the ring of integers of K. Since λ is a prime of O reduction modulo λ defines a map GL(2, O) → GL(2, F) where F = O/λ is the residue class field of λ, and we obtain a mod ℓ Galois representation

\overline{ρ}_{f,λ} : Gal(\overline{Q}/Q) ⟶ GL(2, F).

21.2 Serre’s Conjecture A

Serre [Ser87] conjectured that certain mod ℓ representations arise from modular forms, and then gave a precise recipe for which type of modular form would give rise to the representation. We refer to the first part of his conjecture as “Conjecture A” and to the second as “Conjecture B”.

Let ρ : Gal(\overline{Q}/Q) → GL(2, F) be a Galois representation with F a finite field. We say that ρ arises from a modular form or that ρ is modular if there is some newform f = ∑ a_{n}q^{n} and some prime ideal λ of K = Q(…, a_{n}, …) such that ρ is isomorphic to \overline{ρ}_{f,λ} over \overline{F}.

[figure: O/λ; \overline{F}; F]

Recall that a Galois representation ρ is odd if det(ρ(c)) = −1 where c ∈ Gal(\overline{Q}/Q) is a complex conjugation. Galois representation arising from modular forms are always odd. [More detail?]

Conjecture 21.2.1 (Serre’s Conjecture A). Suppose

ρ : Gal(\overline{Q}/Q) → GL(2, F)

is an odd irreducible (continuous) Galois representation with F a finite field. Then ρ arises from a modular form.

21.2.1 The Field of definition of ρ

One difficulty is that ρ sometimes takes values in a slightly smaller field than O/λ. We illustrate this by way of an example. Let f be one of the two conjugate newforms of level 23, weight 2, and trivial character. Then

f = q + αq^{2} + (−2α − 1)q^{3} + (−α − 1)q^{4} + 2αq^{5} + ⋯

with α^{2} + α − 1 = 0. The coefficients of f lie in O = Z[α] = Z[\frac{1+\sqrt{5}}{2}]. Take λ to be the unique prime of O lying over 2, then O/λ ≅ F_{4} and so \overline{ρ}_{f,λ} is a homomorphism to GL(2, F_{4}).

Proposition 21.2.2. If p ≠ 2 then a_{p} ∈ Z[\sqrt{5}].

Proof. We have f = f_{1} + αf_{2} with

f_{1} = q − q^{3} − q^{4} + ⋯
f_{2} = q^{2} − 2q^{3} − q^{4} + 2q^{5} + ⋯

Because S_{2}(Γ_{0}(23)) has dimension two, it is spanned by f_{1} and f_{2}. Let η(q) = q^{\frac{1}{24}} ∏_{n≥1}(1 − q^{n}). By Proposition ??, g = (η(q)η(q^{23}))^{2} ∈ S_{2}(Γ_{0}(23)). An explicit calculation shows that g = q^{2} − 2q^{3} + ⋯ so we must have g = f_{2}. Next observe that g is a power series in q^{2}, modulo 2:

g = q^{2} ∏(1 − q^{n})^{2}(1 − q^{23n})^{2}
≡ q^{2} ∏(1 − q^{2n})(1 − q^{46n}) (mod 2)
≡ q^{2} ∏(1 + q^{2n} + q^{46n} + q^{48n}) (mod 2)

Thus the coefficient in f_{2} of q^{p} with p ≠ 2 prime is even and the proposition follows.

Thus those a_{p} with p ≠ 2 map modulo λ to F_{2} ⊂ F_{4}, and hence the traces and determinants of Frobenius, at primes where Frobenius is defined, take values in F_{2}. This is enough to imply that \overline{ρ}_{f,λ} is isomorphic over \overline{F} to a representation ρ : Gal(\overline{Q}/Q) → GL(2, F_{2}). Geometrically, GL(2, F_{2}) ≅ S_{3} and the representation ρ is one in which Galois acts via S_{3} on three points of X_{0}(23).

Thus in general, if we start with ρ and wish to see that ρ satisfies Conjecture A, we may need to pass to an algebraic closure of F. In our example, starting with ρ : Gal(\overline{Q}/Q) → GL_{2}(F_{2}) we say that “ρ is modular” because ρ ≅ \overline{ρ}_{λ,f}, but keep in mind that this particular isomorphism only takes place over F_{4}.

At this point K. Buzzard comments, “Maybe there is some better modular form so that all of the a_{p} actually lie in F_{2} and the associated representation is isomorphic to ρ. That would be a stronger conjecture.” Ribet responds that he has never thought about that question.

21.3 Serre’s Conjecture B

Serre’s second conjecture asserts that ρ arises from a modular form in a particular space. Suppose

ρ : Gal(\overline{Q}/Q) → GL(2, F)

is an odd irreducible Galois representation with F a finite field.

Conjecture 21.3.1 (Serre’s Conjecture B). Suppose ρ arises from some modular form. Then ρ arises from a modular form of level N(ρ), weight k(ρ) and character ε(ρ). The exact recipe for N(ρ) and k(ρ) will be given later.

Conjecture B has largely been proven.

Theorem 21.3.2. Suppose ℓ is odd. If the mod ℓ representation ρ is irreducible and modular, then ρ arises from a newform f of level N(ρ) and weight k(ρ).

21.4 The Level

The level N(ρ) is a conductor of ρ, essentially the Artin conductor except that we omit the factor corresponding to ℓ. The level is a product

N(ρ) = ∏_{p≠ℓ} p^{e(p)}.

We now define the e(p). Let V be the representation space of ρ, so V is a two dimensional F-vector space and we view ρ as a homomorphism

ρ : Gal(\overline{Q}/Q) → Aut(V),

or equivalently view V as an F[Gal(\overline{Q}/Q)]-module. Let K = \overline{Q}^{ker(ρ)} be the field cut out by ρ, it is a finite Galois extension of Q with Galois group which we call G. Choose a prime ℘ of K lying over p and let

D = {σ ∈ G : σ(℘) ⊂ ℘}

denote the decomposition group at ℘. Let O be the ring of integers of K and recall that the higher ramification groups G_{−1} ⊃ G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ ⋯ are

G_{i} = {σ ∈ D : σ acts trivially on O/℘^{i+1}}.

Thus G_{−1} = D and G_{0} is the inertia group. Each G_{i} is a normal subgroup of D because G_{i} is the kernel of D → Aut(O/℘^{i+1}). Let

V^{G_{i}} = {v ∈ V : σ(v) = v for all σ ∈ G_{i}}.

Lemma 21.4.1. The subspace V^{G_{i}} is invariant under D.

Proof. Let g ∈ D, h ∈ G_{i} and v ∈ V^{G_{i}}. Since G_{i} is normal in D, there exists h′ ∈ G_{i} so that g^{−1}hg = h′. Then h(gv) = gh′v = gv so gv ∈ V^{G_{i}}.

By [Frö67, §9] that there is an integer i so that G_{i} = 0. We can now define

e(p) = ∑_{i=0}^{∞} \frac{1}{[G_{0} : G_{i}]} dim(V/V^{G_{i}}).

Thus e(p) depends only on ρ|I_{p} where I_{p} = G_{0} is an inertia group at p. In particular, if ρ is unramified at p then all G_{i} for i ≥ 0 vanish and e(p) = 0. Separating out the term dim(V/V^{I_{p}}) corresponding to i = 0 allows us to write

e(p) = dim(V/V^{I_{p}}) + δ(p) ≤ 2 + δ(p)

since dim V = 2. The term δ(p) is called the Swan conductor. By [Ser77, 19.3] δ(p) is an integer, hence e(p) is an integer. We call ρ tamely ramified at p if all G_{i} for i > 0 vanish, in which case the Swan conductor is 0.

Suppose ρ ≅ \overline{ρ}_{f,λ} (over \overline{F}) for some newform f of level N. There is a relationship between e(p) and something involving only f. Let V_{λ} be the representation space of ρ_{f,λ}, so V_{λ} is a vector space over an extension of Q_{ℓ}. It turns out that

ord_{p}(N) = dim(V_{λ}/V_{λ}^{I_{p}}) + δ(p).

Thus

ord_{p}(N) = e(p) + error term

where the error term is dim(V^{I_{p}}) − dim(V_{λ}^{I_{p}}) ≤ 2. The point is that more can become invariant upon reducing modulo λ. Thus if f gives rise to ρ then the power of p in the level N of f is constrained as it is given by a certain formula only depending on e(p) and an error term which has magnitude at most 2.

As we will see, the weight k(ρ) depends only on ρ|I_{ℓ}. Thus k(ρ) can be viewed as an analogue of e(ℓ).

21.4.1 Remark on the case N(ρ) = 1

One consequence of Conjecture B is that every modular mod ℓ representation ρ must come from a newform f of level prime to ℓ. Suppose that f is a modular form of level ℓ. Consider the corresponding mod ℓ representation. It is unramified outside ℓ so N(ℓ) = 1. The conjecture then implies that this mod ℓ representation comes from a level 1 modular form, i.e., a modular form on SL_{2}(Z), of possibly higher weight. This is a classical result.

For the rest of this subsection we assume that ℓ ≥ 11. There is a relationship between mod ℓ forms on SL_{2}(Z) of weight ℓ+1 and mod ℓ forms on Γ_{0}(ℓ) of weight 2. The dimensions of each of these spaces are the same over F_{ℓ} or over C, i.e.,

dim_{F_{ℓ}} S_{2}(Γ_{0}(ℓ); F_{ℓ}) = dim_{C} S_{2}(Γ_{0}(ℓ); C).
dim_{F_{ℓ}} S_{ℓ+1}(SL_{2}(Z); F_{ℓ}) = dim_{C} S_{ℓ+1}(SL_{2}(Z); C).

We now describe a map

F : S_{2}(Γ_{0}(ℓ); F_{ℓ}) ⟶ S_{ℓ+1}(SL_{2}(Z); F_{ℓ}).

The weight k Eisenstein series

E_{k} = 1 − \frac{2k}{B_{k}} ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}

is a modular form for SL_{2}(Z). Here the Bernoulli numbers B_{k} are defined by

\frac{t}{e^{t} − 1} = ∑_{k=0}^{∞} B_{k} \frac{t^{k}}{k!} ,

so B_{0} = 1, B_{1} = −\frac{1}{2}, B_{2} = \frac{1}{12}, B_{3} = −\frac{1}{720}, . . ..

Proposition 21.4.2. If ℓ ≥ 5, then

E_{ℓ−1} ≡ 1 (mod ℓ).

Proof. We must check that ℓ | \frac{2(ℓ−1)}{B_{ℓ−1}}, or equivalently that ℓ | \frac{1}{B_{ℓ−1}} . Set n = ℓ − 1, p = ℓ and apply [Lan95, X.2.2].

Suppose \overline{f} ∈ S_{2}(Γ_{0}(ℓ); F_{ℓ}) is the reduction of f ∈ S_{2}(Γ_{0}(ℓ); F_{ℓ}). Multiplication by E_{ℓ−1} gives a mod ℓ form \overline{f} · E_{ℓ−1} of weight ℓ + 1 on Γ_{0}(ℓ). We have

F (\overline{f}) = tr(\overline{f} · E_{ℓ−1}) ∈ S_{ℓ+1}(SL_{2}(Z); F_{ℓ})

where tr is induced by X_{0}(ℓ) → X_{0}(1). The map F was discovered by Serre [DK73]. That F has the properties necessary to deduce the prime level case of Serre’s Conjecture B was proved by explicit computation in the Berkeley Ph.D. thesis of C. Queen [Que77]. Katz believes that it is easy to prove the formula from the right magical moduli point of view, but neither author has seen this. The first author gave a concrete proof for ℓ > 2 in [Rib94]. It would be nice if someone would construct a clear proof for the case ℓ = 2.

21.4.2 Remark on the proof of Conjecture B

We now give a very brief outline of the proof of Conjecture B when ℓ > 2. Start with a representation ρ which comes from some (possibly terrible) newform f of level N(f) and weight k(f).

Step 1. Using a concrete argument replace f by another newform giving rise to ρ so that ℓ ∤ N(f).

Step 2. Compare N(ρ) and N(f). These are two prime to ℓ numbers. What if p | \frac{N(f)}{N(ρ)}? Carayol separated this into several cases. In each case replace f by a better form so that the ratio has a smaller power of p in it. This is level lowering. Eventually we get to the case N(ρ) = N(f).

Step 3. Using a paper of Edixhoven [Edi92b] one shows that f can be replaced with a another form of the same level so that the weight k is equal to k(ρ).

21.5 The Weight

21.5.1 The Weight modulo ℓ − 1

We first give some background which motivates Serre’s numerical recipe for the weight. We start with a newform f of low weight k and consider the behavior of the representation ρ = \overline{ρ}_{f,λ}|I_{ℓ}, where I_{ℓ} is an inertia group at ℓ. Assume that we are in the following situation:

• N = N(f) is prime to ℓ,

• 2 ≤ k ≤ ℓ + 1.

There are other cases to consider but we consider this one first. Let ε be the character of f and recall that for p ∤ ℓN

det(ρ_{f,ℓ}(Frob_{p})) = ε(p)p^{k−1}.

The mod ℓ cyclotomic character χ_{ℓ} : Gal(\overline{Q}/Q) → F_{ℓ}^{∗} is the homomorphism sending Frob_{p} to p ∈ F_{ℓ}^{∗}. Thus

det(ρ) = εχ_{ℓ}^{k−1}.

Since ε is a Dirichlet character mod N and ℓ ∤ N, ε|I_{ℓ} = 1 and so

det(ρ|I_{ℓ}) = χ_{ℓ}^{k−1}.

We see immediately that det(ρ|I_{ℓ}) determines k modulo ℓ − 1. Since 2 ≡ ℓ + 1 (mod ℓ − 1), this still doesn’t distinguish between 2 and ℓ + 1.

21.5.2 Tameness at ℓ

Let

ρ : Gal(\overline{Q}/Q) ⟶ GL(2, F_{ℓ^{ν}})

be a modular mod ℓ Galois representation. Let D = D_{ℓ} ⊂ Gal(\overline{Q}/Q) be a decomposition group at ℓ. Let σ be the semisimplification of ρ|D. Thus σ is either a direct sum of two characters or σ = ρ|D depending on whether or not ρ|D is irreducible. By [Frö67, 8.1] the ramification group P = G_{1} is the unique Sylow ℓ-subgroup of I_{ℓ} = G_{0}.

Lemma 21.5.1. The semisimplification σ of ρ is tame, i.e., σ|P = 0 where P is the Sylow ℓ-subgroup of the inertia group I_{ℓ}.

Proof. Since I_{ℓ} is normal in D and any automorphism of I_{ℓ} sends P to some Sylow ℓ-subgroup and P is the only such, it follows that P is normal in D. Let W = F_{ℓ^{ν}} × F_{ℓ^{ν}} be the representation space of σ. Then

W^{P} = {w ∈ W : σ(τ)w = w for all τ ∈ P }

is a subspace of W invariant under the action of D. For this let α ∈ D and suppose w ∈ W^{P}. Since P is normal in D, α^{−1}τα = τ′ for some τ′ ∈ P. Therefore

σ(α)^{−1}σ(τ)σ(α)w = σ(τ′)w = w

so σ(τ)σ(α)w = σ(α)w hence σ(α)w ∈ W^{P}.

But W^{P} ≠ 0. To see this write W as a disjoint union of its orbits under the action of P. Since P is an ℓ-Sylow group and W is finite we see that the size of each orbit is either 1 or a positive power of ℓ. Now {0} is a singleton orbit, W has ℓ-power order, and all non-singleton orbits have order a positive power of ℓ so there must be at least ℓ − 1 other singleton orbits. Each of these other singleton orbits gives a nonzero element of W^{P}.

If W^{P} = W then P acts trivially so we are done. If W^{P} ≠ W, then since W^{P} is nonzero it is a one dimensional subspace invariant under D, so by semisimplicity σ is diagonal. Let τ ∈ P, then τ has order ℓ^{n} for some n. Write

σ(τ) = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ,

then α^{ℓ^{n}} = β^{ℓ^{n}} = 1. Since α, β ∈ F_{ℓ^{ν}} they have order dividing |F_{ℓ^{ν}}^{∗} | = ℓ^{ν} − 1. But gcd(ℓ^{ν} − 1, ℓ^{n}) = 1 from which it follows that α = β = 1. Thus P = {1} and again P acts trivially, as claimed.

21.5.3 Fundamental characters of the tame extension

The lemma implies σ|I_{ℓ} factors through the tame quotient I_{t} = I_{ℓ}/P. We now describe certain characters of I_{t} more explicitly. Denote by Q_{ℓ}^{tm} the maximal tamely ramified extension of Q_{ℓ}, it is the fixed field of P. Let K = Q_{ℓ}^{ur} be the maximal unramified extension, i.e., the fixed field of the inertia group I_{ℓ}. By Galois theory

I_{t} = Gal(Q_{ℓ}^{tm}/Q_{ℓ}^{ur}).

For each positive integer n coprime to ℓ there is a tower of Galois extensions

Q_{ℓ}^{tm} I_{t} K( \sqrt[n]{ℓ}) μ_{n}(K) K = Q_{ℓ}^{ur} Q_{ℓ}

Since Q(ζ_{n})/Q is unramified at ℓ the nth roots of 1 are contained in K so by Kummer theory

Gal(K( \sqrt[n]{ℓ})/K) ≅ μ_{n}(K)

where μ_{n}(K) denotes the group of nth roots of unity in K. Thus for each n prime to ℓ we obtain a map I_{t} → μ_{n}(K). They are compatible so upon passing to the limit we obtain a map

I_{t} ⟶ \underleftarrow{lim} μ_{n}(K) = ∏_{r≠ℓ} \underleftarrow{lim} μ_{r^{a}}(K) = ∏_{r≠ℓ} Z_{r}(1).

In fact [Frö67, 8, Corollary 3] the above maps are isomorphisms. Viewed more cleverly mod ℓ we obtain a map

I_{t} ⟶ \underleftarrow{lim} μ_{n}(\overline{F}_{ℓ}) = \underleftarrow{lim} F_{ℓ^{i}}^{∗}.

Thus for each i we have a map

I_{t} → F_{ℓ^{i}}^{∗}.

which is called the fundamental character of level i. The construction of this character on I_{t} is somewhat unnatural because we had to choose an embedding F_{ℓ^{i}} ↪ \overline{F}_{ℓ}. Instead Serre begins with a “disembodied” field F of order ℓ^{i}. There are i different maps F_{ℓ^{i}} → F corresponding to the i automorphisms of F_{ℓ^{i}}. Restricting these maps to F_{ℓ^{i}}^{∗} and composing with I_{t} → F_{ℓ^{i}}^{∗} gives the i fundamental characters of level i. The unique fundamental character of level 1 is the mod ℓ cyclotomic character χ_{ℓ}.

21.5.4 The Pair of characters associated to ρ

Recall that we have a representation ρ whose semisimplification gives a representation which we denote by σ:

σ : I_{t} ⟶ GL(2, F_{ℓ^{ν}}).

Since σ(I_{t}) is a finite abelian group and the elements of I_{t} have order prime to ℓ, this representation is semisimple and can be diagonalized upon passing to \overline{F}_{ℓ}. In fact, since the characteristic polynomials all have degree two, σ can be diagonalized over F_{ℓ^{2ν}}. Thus σ corresponds to a pair of characters

α, β : I_{t} ⟶ F_{ℓ^{2ν}}^{∗} .

These characters have some stability properties since σ is the restriction of a homomorphism from the full decomposition group. Consider the tower of fields

K( \sqrt[n]{ℓ})

K = Q_{ℓ}^{ur}
Q_{ℓ}

Let G = Gal(K( \sqrt[n]{ℓ})/Q_{ℓ}). Recall that Gal(K( \sqrt[n]{ℓ})/K) ≅ μ_{n}(K) and Gal(K/Q_{ℓ}) is topologically generated by Frob_{ℓ}. If h ∈ μ_{n}(K) and g ∈ G restricts to Frob_{ℓ} then we have the conjugation formula:

ghg^{−1} = h^{ℓ}.

Applying this to our representation σ with h ∈ I_{t} we find that

σ(ghg^{−1}) = σ(h^{ℓ}) = σ(h)^{ℓ}

so

σ(g)σ(h)σ(g^{−1}) = σ(ghg^{−1}) = σ(h)^{ℓ}.

The point is that the representation h ↦ σ(h)^{ℓ} is equivalent to h ↦ σ(h) via conjugation by σ(g). We conclude that the pair of characters {α, β} is stable under ℓ-th powering, i.e., as a set

{α, β} = {α^{ℓ}, β^{ℓ}}.

What does this mean? There are two possibilities:

• Level 1: α^{ℓ} = α and β^{ℓ} = β.

• Level 2: α^{ℓ} = β and β^{ℓ} = α, but α ≠ β.

Note that in the level 1 case, α and β take values in F_{ℓ^{ν}}^{∗}.

21.5.5 Recipe for the weight

We will play a carnival game, “guess your weight.” First we consider the level 2 case. Our strategy is to express α and β in terms of the two fundamental characters of level 2. We then observe how the characters associated to a newform are expressed in terms of the two fundamental characters.

The remainder of this chapter is devoted to motivating the following definition.

Definition 21.5.2. Let ρ and σ be as above. Let ψ, ψ′ be the two fundamental characters of level 2 and χ the fundamental character of level 1 (the cyclotomic character). Serre’s recipe for k(ρ) is as follows.

1. Suppose that α and β are of level 2. We have

ρ|I_{ℓ} = (\begin{matrix}α & 0 \\ 0 & β \end{matrix}) .

After interchanging α and β if necessary, we have (uniquely) α = ψ^{r+ℓq} = ψ^{r}(ψ′)^{q} and β = (ψ′)^{r}ψ^{q} with 0 ≤ r < q ≤ ℓ − 1. We set k(ρ) = 1 + ℓr + q.

2. Suppose that α and β are of level 1. We have

ρ|I_{ℓ} = (\begin{matrix}χ^{r} & ∗ \\ 0 & χ^{q} \end{matrix}) .

(a) If ∗ = 0, normalize and reorder r, q so that 0 ≤ r ≤ q ≤ ℓ − 2. We set k(ρ) = 1 + ℓr + q.

(b) If ∗ ≠ 0, normalize so that 0 ≤ q ≤ ℓ − 2 and 1 ≤ r ≤ ℓ − 1. We set a = min(r, q), b = max(r, q). If χ^{r−q} = χ and ρ ⊗ χ^{−q} is not finite at ℓ then we set k(ρ) = 1 + ℓa + b + ℓ − 1; otherwise we set k(ρ) = 1 + ℓa + b.

21.5.6 The World’s first view of fundamental characters

First some background. Suppose E/Q is an elliptic curve and ℓ is a prime (2 is allowed). Assume E has good supersingular reduction at ℓ, so Ẽ(\overline{F}_{ℓ})[ℓ] = \{0\}. Then there is a Galois representation

ρ : Gal(\overline{Q}/Q) ⟶ Aut(E[ℓ])

which, a priori, may or may not be irreducible. As above, ρ gives rise to two characters

α, β : I_{t} → F_{ℓ^{2}}^{∗}.

Serre (see [V´72]) proved that α, β are the two fundamental characters of level 2 and that I_{t} acts irreducibly over F_{ℓ}. [[Is this right?]] He also observed that there is a map

I_{t} → F_{ℓ^{2}}^{∗} ⊂ GL(2, F_{ℓ})

where F_{ℓ^{2}}^{∗} sits inside GL(2, F_{ℓ}) via the action of the multiplicative group of a field on itself after choice of a basis, and the map to F_{ℓ^{2}}^{∗} is through one of the two fundamental characters of level 2.

21.5.7 Fontaine’s theorem

Serre next asked Fontaine to identify the characters α and β attached to more general modular representations. Fontaine’s answer was published in [Edi92b]. Suppose f = ∑ a_{n}q^{n} is a newform of weight k such that 2 ≤ k ≤ ℓ and the level N(f) of f is prime to ℓ. Let ρ = \overline{ρ}_{f,λ} where λ is a prime of Q(…, a_{n}, …) lying over ℓ. Assume we are in the supersingular case, i.e.,

a_{ℓ} ≡ 0 mod λ.

Semisimplifying as before gives a pair of characters

α, β : I_{t} ⟶ F_{ℓ^{2ν}}^{∗} .

Let ψ, ψ′ : I_{t} → F_{ℓ^{2}}^{∗} be the two fundamental characters of level 2 so ψ = (ψ′)^{ℓ} and ψ′ = ψ^{ℓ}.

Theorem 21.5.3. With the above notation and hypothesis, the characters α, β arising by semisimplifying and restricting \overline{ρ}_{f,λ} satisfy

{α, β} = {ψ^{k−1}, (ψ′)^{k−1}}.

21.5.8 Guessing the weight (level 2 case)

We now try to guess the weight in the level 2 case. We begin with a representation ρ whose semisimplification is a representation σ, which in turn gives rise to a pair of level 2 characters

{α, β} = {ψ^{a}, (ψ′)^{a} = ψ^{ℓa}}.

Since ψ takes values in F_{ℓ^{2}}^{∗}, we may think of a as a number modulo ℓ^{2} − 1. Note also that the pair is unchanged upon replacing a by ℓa. The condition that we are not in level 1 is that a is not divisible by ℓ + 1 since

ψψ′ = ψ^{ℓ+1} : I_{t} → F_{ℓ}^{∗}

is the unique fundamental character of level 1, i.e., the mod ℓ cyclotomic character.

Normalize a so that 0 ≤ a < ℓ^{2} − 1 and write a = qℓ + r. What are the possible values for q and r? By the Euclidean algorithm 0 ≤ r ≤ ℓ − 1 and 0 ≤ q ≤ ℓ − 1. If r = q then a is a multiple of ℓ + 1, so r ≠ q. If r > q, multiply the above relation by ℓ to obtain aℓ = qℓ^{2} + rℓ. But we are working mod ℓ^{2} − 1 so this becomes aℓ = q + rℓ (mod ℓ^{2} − 1). Thus if we replace a by aℓ then the roles of q and r are swapped in the Euclidean division. Thus we can assume that 0 ≤ r < q ≤ ℓ − 1. Now

α = ψ^{a} = ψ^{qℓ+r} = (ψ′)^{q}ψ^{r} = (ψψ′)^{r}(ψ′)^{q−r},

so

{α, β} = {(ψψ′)^{r}(ψ′)^{q−r}, (ψψ′)^{r}ψ^{q−r}}.

Since ψψ′ = χ is the mod ℓ cyclotomic character, we can view {α, β} as a pair of characters (ψ′)^{q−r} and ψ^{q−r} which has been multiplied, as a pair, by χ^{r}.

What weight do we guess if r = 0? In this case

{α, β} = {(ψ′)^{k−1}, ψ^{k−1}}

where k = 1 + q. So in analogy with Theorem 21.5.3 we guess that

k(ρ) = 1 + q (r = 0, supersingular case).

What do we guess in general? Suppose f = ∑ a_{n}q^{n} is a modular form thought of mod ℓ which gives rise to ρ, and that ℓ does not divide the level of f. We might as well ask what modular form gives rise to ρ ⊗ χ. In [?] we learn that

θf = ∑ na_{n}q^{n} (mod ℓ)

is a mod ℓ eigenform, and it evidently gives rise to ρ ⊗ χ. Furthermore, if k is the weight of f, then θf has weight k + ℓ + 1. Since

{α, β} = {ψ^{q−r}, (ψ′)^{q−r}} · χ^{r}

we guess that

k(ρ) = q − r + 1 + (ℓ + 1)r = 1 + ℓr + q (supersingular case)

But be careful! the minimal weight k does not have to go up by ℓ − 1, though it usually does. This is described by the theory of θ-cycles which we will review shortly.

21.5.9 θ-cycles

The theory of the θ operator was first developed by Serre and Swinnerton–Dyer and then later jazzed up by Katz in [?]. There is a notion of modular forms mod ℓ and of q-expansion which gives a map

α : ⨁_{k≥0} M_{k}(Γ_{1}(N); F_{ℓ}) ⟶ F_{ℓ}[[q]].

This map is not injective. The kernel is the ideal generated by A − 1 where A is the Hasse invariant.

Suppose f ∈ F_{ℓ}[[q]] is in the image of α. If f ≠ 0 let w(f) denote the smallest k so that f comes from some M_{k}. If f does not coming from any single M_{k} do not define w(f). Define an operator θ on F_{ℓ}[[q]] by

θ(∑ a_{n}q^{n}) = q \frac{d}{dq} (∑ a_{n}q^{n}) = ∑ na_{n}q^{n}.

Serre and Swinnerton-Dyer showed that θ preserves the image of α.

Theorem 21.5.4. Suppose f ≠ 0 is a mod ℓ modular form as above. If ℓ ∤ w(f) then w(θf) = w(f) + ℓ + 1.

Associated to f we have a sequence of nonnegative integers

w(f), w(θf), w(θ^{2}f), …

Fermat’s little theorem implies that this sequence is periodic because θ^{ℓ}f = θf and so w(θ^{ℓ}f) = w(θf). We thus call the cyclic sequence w(f), w(θf), w(θ^{2}f), … the θ-cycle of f. Tate asked:

What are the possible θ-cycles?

This question was answered in [Joc82] and [Edi92b]. We now discuss the answer in a special case.

Let f be an eigenform such that 2 ≤ k = w(f) ≤ ℓ and f is supersingular, i.e., a_{ℓ}(f) = 0. Since f is an eigenform the a_{n} are multiplicative (a_{nm} = a_{n}a_{m} for (n, m) = 1) and if ε denotes the character of f then

a_{ℓ^{i}} = a_{ℓ^{i−1}} · a_{ℓ} − ε(ℓ)ℓ^{k−1}a_{ℓ−2}
= a_{ℓ^{i−1}} · a_{ℓ} = 0

since we are working in characteristic ℓ and k ≥ 2. Thus a_{n}(f) = 0 whenever ℓ | n and so θ^{ℓ−1}f = f hence w(θ^{ℓ−1}f) = w(f).

If we apply θ successively to f what happens? Before proceeding we remark that it can be proved that there is at most one drop in the sequence

k = w(f), w(θf), w(θ^{2}f), …, w(θ^{ℓ−2}f).

First suppose k = 2. The θ-cycle must be

2, 2 + (ℓ + 1), 2 + 2(ℓ + 1), …, 2 + (ℓ − 2)(ℓ + 1).

This is because, by Theorem 21.5.4, applying θ raises the weight by ℓ + 1 so long as the weight is not a multiple of ℓ. Only the last term in the above sequence is divisible by ℓ. There are ℓ − 1 terms so this is the full θ-cycle.

Next suppose k = ℓ. The θ-cycle is

ℓ, 3, 3 + (ℓ + 1), …, 3 + (ℓ + 1)(ℓ − 3).

The last term is divisible by ℓ, no earlier term after the first is, and there are ℓ − 1 terms so this is the full θ-cycle. We know that the second term must be 3 since it is the only number so that the θ-cycle works out right, i.e., so that the (ℓ − 1)st term is divisible by ℓ but no earlier term except the first is. For example, if we would have tried 2 instead of 3 we would have obtained the sequence

ℓ, 2, 2 + (ℓ + 1), …, 2 + (ℓ + 1)(ℓ − 3), 2 + (ℓ + 1)(ℓ − 2).

This sequence has one too many terms.

Now we consider the remaining values of k: 2 < k < ℓ. The θ-cycle is

k, k + (ℓ + 1), …, k + (ℓ + 1)(ℓ − k), k′, k′ + ℓ + 1, …, k′ + (ℓ + 1)(k − 3).

The first ℓ − k + 1 terms of the sequence are obtained by adding ℓ + 1 successively until obtaining a term k + (ℓ + 1)(ℓ − k) which is divisible by ℓ. Applying θ to a form of weight k + (ℓ + 1)(ℓ − k) causes the weight to drop to some k′. How can we guess k′? It must be such that k′ + (ℓ + 1)(k − 3) is divisible by ℓ. Thus the correct answer is

k′ = ℓ + 3 − k.

21.5.10 Edixhoven’s paper

Suppose that ρ is an irreducible mod ℓ representation so that the level of the characters α, β associated to the semisimplification of ρ are level 2. To avoid problems in a certain exceptional case assume ℓ ≥ 3. Edixhoven [Edi92b] proved that if ρ arises from an eigenform in S_{k}(Γ_{1}(N)) with (N, ℓ) = 1, then ρ arises from an eigenform in S_{k(ρ)}(Γ_{1}(N)). What are the elements of the proof?

1. The behavior of ρ|I_{ℓ} when ρ arises from an f with 2 ≤ w(f) ≤ ℓ + 1.

2. The fact that every modular representation ρ has the form \overline{ρ}_{f,λ} ⊗ χ^{i} where i ∈ Z/(ℓ − 1)Z, and 2 ≤ w(f) ≤ ℓ + 1.

Serre knew the second element but never published a proof. How did Serre talk about his result before Edixhoven’s paper? The eigenform f correponds to a system of eigenvalues in \overline{F}_{ℓ} of the Hecke operators T_{r}, r ∤ ℓN. Eigenforms of weight at most ℓ + 1 give, up to twist, all systems of Hecke eigenvalues. A possible proof of this uses the construction of ρ_{f} in terms of certain étale cohomology groups.

21.6 The Character

Let

ρ : G_{Q} = Gal(\overline{Q}/Q) ⟶ GL(2, F_{ℓ^{ν}})

be a Galois representation. Assume that ρ is irreducible, modular ρ ≅ \overline{ρ}_{λ,f}, and ℓ > 2. The degree of a character φ : (Z/NZ)^{∗} → C^{∗} is the cardinality |φ((Z/NZ)^{∗})| of its image.

Theorem 21.6.1. Under the above hypothesis, ρ comes from a modular form f of weight k(ρ), level N(ρ), and under a certain extra assumption, character ε of degree prime to ℓ.

Extra Assumption: Not all of the following are true.

1. ℓ = 3, 2. ρ| Gal(\overline{Q}/Q(\sqrt{−3})) is abelian, and

3. det(ρ) is not a power of the mod 3 cyclotomic character χ.

Example 21.6.2. If ρ comes from the Galois representation on the 3-torsion of an elliptic curve, then det(ρ) = χ is the mod 3 cyclotomic character, so the extra assumption does hold and the theorem applies.

Let ρ be a representation as above. Then it is likely that

det ρ : G_{Q} → F_{ℓ^{ν}}

is ramified at ℓ. Let χ be the mod ℓ cyclotomic character.

Proposition 21.6.3. Let φ : G_{Q} → F_{ℓ^{ν}}^{∗} be a continuous homomorphism. Then φ = θχ^{i} for some i and some θ : G_{Q} → F_{ℓ^{ν}}^{∗} which is unramified at ℓ.

Proof. Since φ is continuous and F_{ℓ^{ν}}^{∗} is finite, the subfield K of \overline{Q} fixed by ker(φ) is a finite Galois extension of Q. Since the image of φ is abelian, the Galois group of K over Q is abelian. By the Kronecker-Weber theorem [Lan94, X.3] there is a cyclotomic extension Q(ζ_{n}) = Q(exp^{2πi/n}) which contains K. Since Gal(Q(ζ_{n})/Q) ≅ (Z/nZ)^{∗}, the character φ gives a homomorphism φ′ : (Z/nZ)^{∗} → F_{ℓ^{ν}}^{∗}. Write n = mℓ^{j} with m coprime to ℓ. Then

(Z/nZ)^{∗} ≅ (Z/mZ × Z/ℓ^{j}Z)^{∗} ≅ (Z/mZ)^{∗} × (Z/ℓ^{j}Z)^{∗}

which decomposes φ′ as a product θ′ × ψ′ where

θ′ : (Z/mZ)^{∗} → F_{ℓ^{ν}}^{∗}
ψ′ : (Z/ℓ^{j}Z)^{∗} → F_{ℓ^{ν}}^{∗} .
[figure: \overline{Q}; Q(ζ_{n}); Q(ζ_{ℓ^{j}}); (Z/ℓ^{j}Z)^{∗}; Q(ζ_{m}); (Z/mZ)^{∗}; K; ⊂F_{ℓ^{ν}}^{∗}; Q]

The character θ is obtained by lifting θ′. It is unramified at ℓ because it factors through Gal(Q(ζ_{m})/Q) and ℓ ∤ m. The cardinality of (Z/ℓ^{j}Z)^{∗} is (ℓ − 1)ℓ^{j−1} whereas the cardinality of F_{ℓ^{ν}}^{∗} is (ℓ − 1)(ℓ^{ν−1} + ⋯ + 1) so the image of ψ′ lies in F_{ℓ}^{∗}. Thus ψ′ lifts to a power χ^{i} of the cyclotomic character.

Using the proposition write det(ρ) = θχ^{i} with θ unramified at ℓ. As in the proof of the proposition we can write θ as a Dirichlet character (Z/mZ)^{∗} → F_{ℓ^{ν}}^{∗}. It can be shown using properties of conductors that m can be chosen so that m|N(ρ). Thus we view θ as a Dirichlet character

θ : (Z/N(ρ)Z)^{∗} → F_{ℓ^{ν}}^{∗} .

Let H = ker θ ⊂ (Z/N(ρ)Z)^{∗}. Define a congruence subgroup Γ_{H}(N) by

Γ_{H}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N) : a, d ∈ H\} .

We have the following theorem.

Theorem 21.6.4. Suppose ℓ > 2 and ρ satisfies the above assumptions including the extra assumption. Then ρ arises from a form in

S_{k(ρ)}( Γ_{H}(N(ρ)) ).

In particular, the theorem applies if ρ comes from the ℓ-torsion representation on an elliptic curve, since then det = χ so the extra assumption is satisfied.

21.6.1 A Counterexample

One might ask if the extra assumption in Theorem 21.6.4 is really necessary. At first Serre suspected it was not. But he was surprised to discover the following example which shows that the extra assumption can not be completely eliminated. The space S_{2}(Γ_{1}(13)) is 2 dimensional, spanned by the eigenform

f = q + αq^{2} + (−2α − 4)q^{3} + (−3α − 7)q^{4} + (2α + 3)q^{5} + ⋯

and the Gal(Q(\sqrt{−3})/Q)-conjugate of f, where α^{2} + 3α + 3 = 0. The character ε : (Z/13Z)^{∗} → C^{∗} of f has degree 6. Let λ = (\sqrt{3}) and let

\overline{ρ}_{f,λ} : Gal(\overline{Q}/Q) → GL(2, F_{3})

be the associated Galois representation. Then det(\overline{ρ}_{f,λ}) = χθ where χ is the mod 3 cyclotomic character and θ ≡ ε (mod 3). In particular θ has order 2. Thus H = ker(θ) ⊂ (Z/13Z)^{∗} is exactly the index two subgroup of squares in (Z/13Z)^{∗}. The conclusion of the theorem can not hold since S_{2}(Γ_{H}(13)) = 0. This is because any form would have to have a character whose order is at most two since it must be trivial on H, but S_{2}(Γ_{H}(13)) ⊂ S_{2}(Γ_{1}(13)) and S_{2}(Γ_{1}(13)) is spanned by f and its Galois conjugate, both of which have character of order 6. In this example

• ℓ = 3,

• \overline{ρ}_{λ,f}| Gal(\overline{Q}/Q(\sqrt{−3})) is abelian, and

• det ρ is not a power of χ.

A good way to see the second assertion is to consider the following formula:

f ⊗ ε^{−1} = \overline{f}

(up to an Euler factor at 13) in the sense that

ρ_{f,λ} ⊗ ε^{−1} = ρ\overline{f}.

Now reduce mod 3 to obtain

\overline{ρ}_{f,λ} ⊗ ε^{−1} ≅ \overline{ρ}_{f,λ}

since f ≡ \overline{f} (mod \sqrt{−3}) (since 3 is ramified). Thus \overline{ρ}_{f,λ} is isomorphic to a twist of itself by a complex character so \overline{ρ}_{f,λ} is reducible and abelian over the field corresponding to its kernel. In fact, by the same argument, \overline{ρ}_{f,λ} is also abelian when restricted to the Galois groups of Q(\sqrt{13}) and Q(\sqrt{39}).

[[Give more details and describe David Jones’s thesis.]]

21.7 The Weight revisited: level 1 case

We are interested in the recipe for k(ρ) in the level 1 case. If we semisimplify and restrict to inertia we obtain a direct sum of two representations. In the level 1 case both representations are powers of the cyclotomic character. There are thus two possibilities for ρ|I:

ρ|I = (\begin{matrix}χ^{α} & ∗ \\ 0 & χ^{β} \end{matrix}) or ρ|I = (\begin{matrix}χ^{α} & 0 \\ 0 & χ^{β} \end{matrix}) .

In the second case we guess k(ρ) by looking at the exponents and normalizing as best we can. Since α and β are only defined mod ℓ − 1 we may, after relabeling if necessary, assume that 0 ≤ α ≤ β ≤ ℓ − 2. Factoring out χ^{α} we obtain

χ^{α} ⊗ (\begin{matrix}1 & 0 \\ 0 & χ^{β−α} \end{matrix}) .

Next (secretly) recall that if f is ordinary of weight k then f gives rise to the representation

(\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix})

(here ∗ can be trivial). If f is of weight β − α + 1, applying the θ operator α times gives the desired representation. Thus the recipe for the weight is

w(ρ) = (ℓ + 1)α + β − α + 1 = β + ℓα + 1.

There is one caveat: Serre was uncomfortable with weight 1 forms, so if α = β = 0 he defines k(ρ) = ℓ instead of k(ρ) = 1.

Ogus asks what is wrong with weight 1, and Ribet replies that Serre didn’t know a satisfactory way in which to define modular forms in weight 1. Merel then adds that Serre was frustrated because he could not do explicit computations in weight 1.

21.7.1 Companion forms

Suppose f is ordinary (a_{ℓ} ∉ λ) of weight k, 2 ≤ k ≤ ℓ + 1, and let \overline{ρ}_{f,λ} be the associated representation. Then

\overline{ρ}_{f,λ}|I_{ℓ} = (\begin{matrix}χ^{k−1} & ∗ \\ 0 & 1 \end{matrix}) .

To introduce the idea of a companion form suppose that somehow by chance ∗ = 0. Twisting ρ by χ^{1−k} gives

ρ ⊗ χ^{1−k}|I_{ℓ} = ρ ⊗ χ^{ℓ−k}|I_{ℓ} = (\begin{matrix}1 & 0 \\ 0 & χ^{ℓ−k} \end{matrix}) .

What is the minimum weight of a newform giving rise to such a representation? Because the two characters 1 and χ take values in F_{ℓ}^{∗} we are in the level 1 situation. The representation is semisimple, reducible, and α = 0, β = ℓ − k, so the natural weight is ℓ+1−k. Thus Conjecture B predicts that there should exist another form g of weight ℓ + 1 − k such that ρ_{g} ≅ ρ_{f} ⊗ χ^{ℓ−k} (over \overline{F}_{ℓ}). Such a form g is called a companion of f. In characteristic ℓ, we have g = θ^{ℓ−k}f. This conjecture was for the most part proved by Gross [Gro90] when k ≠ 2, ℓ, and by Coleman-Voloch [CV92] when k = ℓ. [[It would be nice to say something here about the subtleties involved in going from this mod ℓ form g to the companion form g produced by Gross-Coleman-Voloch. Roughly, how are the two objects linked?]]

21.7.2 The Weight: the remaining level 1 case

Let

ρ : Gal(\overline{Q}/Q) ⟶ GL(2, F_{ℓ^{ν}})

be a Galois representation with ℓ > 2. Assume that ρ is irreducible and modular. Then ρ comes from a modular form in S_{k(ρ)}(Γ_{1}(Nℓ)), with 2 ≤ k(ρ) ≤ ℓ^{2} − 1. We still must define k(ρ) in the remaining level 1 case in which

ρ|I_{ℓ} = (\begin{matrix}χ^{α} & ∗ \\ 0 & χ^{β} \end{matrix}) .

If α ≠ β + 1 then

k(ρ) = 1 + ℓa + b

where a = min(α, β) and b = max(α, β). Now assume α = β + 1. Then

ρ|I_{ℓ} = χ^{β} ⊗ (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix}) .

Define a representation of Gal(\overline{Q}/Q) by

σ = ρ ⊗ χ^{−β} ≅ (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix}) .

We now give a motivated recipe for k(σ). Granted that “finite at ℓ” is defined in the next section, the recipe is

k(σ) = \begin{cases}2 & if σ is finite at ℓ, \\ ℓ + 1 & otherwise. \end{cases}

This is enough to determine k(ρ) giving

k(ρ) = k(χ^{β} ⊗ σ) = (ℓ + 1)β + k(σ).

21.7.3 Finiteness

We continue with the notation of the previous section. Let

D = D_{ℓ} = Gal(\overline{Q}_{ℓ}/Q_{ℓ}) ↪ Gal(\overline{Q}/Q)

denote a decomposition group at ℓ.

Definition 21.7.1. We say that σ|D is finite if it is equivalent to a representation of the form G(\overline{Q}_{ℓ}), where G is a finite flat F_{ℓ^{ν}} -vector space scheme over Z_{ℓ}.

This definition may not be terribly enlightening, so we consider the following special case. Suppose E/Q is an elliptic curve with semistable (=good or multiplicative) reduction at ℓ. Then E[ℓ] defines a representation

σ : Gal(\overline{Q}/Q) → Aut E[ℓ] ≅ GL(2, F_{ℓ}).

Let Δ_{E} be the minimal discriminant of E.

Proposition 21.7.2. With notation as above, σ is finite at ℓ if and only if

ord_{ℓ} Δ_{E} ≡ 0 (mod ℓ).

If p ≠ ℓ (and E has semistable reduction at p) then σ is unramified at p if and only if

ord_{p} Δ_{E} ≡ 0 (mod ℓ).

We give some hint as to how the proof goes when p ≡ 1 (mod ℓ). Let E be an elliptic curve with multiplicative reduction at p. Set V = E[ℓ] = F_{ℓ} ⊕ F_{ℓ}. We have a representation

σ : D = Gal(\overline{Q}_{ℓ}/Q_{ℓ}) → Aut V.

The theory of Tate curves gives an exact sequence of D-modules

0 → X → V ⟶^{α} Y → 0.

Each of these terms is a D module and X and Y are 1-dimensional as F_{ℓ}-vector spaces. The action of D on Y is given by an unramified character ε of degree dividing 2. The action of D on X is given by χε.

Next we define an element of H^{1}(D, Hom_{F_{ℓ}}(Y, X)). A splitting s : Y → V is an F_{ℓ}-linear map (not necessarily a map of D-modules) such that αs = 1. Choose such a splitting. For each d ∈ D consider the twisting ^{d}s : Y → V defined by ds(y) = ds(d^{−1}y). Since

α(ds(d^{−1}y)) = d(α(s(d^{−1}y))) = d(d^{−1}y) = y

if follows that ^{d}s is again a splitting. Thus ^{d}s − s : Y → V followed by α : V → Y is the zero map. Since ^{d}s − s is a linear map, ^{d}s − s ∈ Hom_{F_{ℓ}}(Y, X). The map d ↦ ^{d}s defines a 1-cocycle which gives an element of H^{1}(D, Hom_{F_{ℓ}}(Y, X)). There is an isomorphism of D-modules Hom_{F_{ℓ}}(Y, X) ≅ μ_{ℓ} so we have isomorphisms

H^{1}(D, Hom_{F_{ℓ}}(Y, X)) ≅ H^{1}(D, μ_{ℓ}) ≅ Q_{p}^{∗}/(Q_{p}^{∗})^{ℓ}.

The last isomorphism follows from Kummer theory since Q_{p} is assumed to contain the ℓ-th roots of unity (our assumption that p ≡ 1 (mod ℓ)). Thus σ defines an element of Q_{p}^{∗}/(Q_{p}^{∗})^{ℓ}. Serre proved that τ is finite if and only if the corresponding element of Q_{p}^{∗}/(Q_{p}^{∗})^{ℓ} is in the image of Z_{p}^{∗}.

If E/Q_{p} is an elliptic curve with multiplicative reduction then there exists a Tate parameter q ∈ Q_{p}^{∗} with Val_{p}(q) > 0 such that

E ≅ E_{q} := G_{m}/q^{Z}

over the unique quadratic unramified extension of Q_{p}. The kernel of multiplication by ℓ gives rise to an exact sequence as above, which is obtained by applying the snake lemma connecting E[ℓ] to Y in the following diagram:

\xymatrix{0 \ar[r] \ar[d] & q^{Z} \ar[r]^{ℓ} \ar[d] & q^{Z} \ar[r] \ar[d] & Y \ar[d] \\ X \ar[d] \ar[r] & G_{m} \ar[r]^{ℓ} \ar[d] & G_{m} \ar[d] \ar[r] & 0 \\ E[ℓ] \ar[r] & E \ar[r]^{ℓ} & E}

Furthermore, the element of Q_{p}^{∗}/(Q_{p}^{∗})^{ℓ} defined by the representation comming from E[ℓ] is just the image of q. One has

Δ_{E} = q ∏_{n=1}^{∞}(1 − q^{n})^{24}.

Note that the product factor is a unit in Q_{p}, so Val_{p} Δ_{E} = Val_{p} q.

Fermat’s Last Theorem

22.1 The application to Fermat

“As part of this parcel, I can sketch the application to Fermat.”

Suppose that semistable elliptic curves over Q are modular. Then FLT is true. Why? “As I have explained so many times… ” Suppose ℓ > 5 and

a^{ℓ} + b^{ℓ} + c^{ℓ} = 0

with abc ≠ 0, all relatively prime, and such that A = a^{ℓ} ≡ −1 (mod 4), B = b^{ℓ} is even and C = c^{ℓ} ≡ 1 (mod 4). Then we consider the elliptic curve

E : y^{2} = x(x − A)(x − B).

The minimal discriminant is

Δ_{E} = \frac{(ABC)^{2}}{2^{8}}

as discussed in Serre’s [Ser87] and [DK95]. The conductor N_{E} is equal to the product of the primes dividing ABC (so in particular N_{E} is square-free). Furthermore, E is semistable – the only hard place to check is at 2. Diamond-Kramer checks this explicitly.

Here is how to get Fermat’s theorem. View E[ℓ] as a G = Gal(\overline{Q}/Q)-module. The idea is to show that this representation must come from a modular form of weight 2 and level 2. This will be a contradiction since there are no modular forms of weight 2 and level 2. But to apply the level and weight theorem we need to know that E[ℓ] is irreducible. The proof of this is due to Mazur.

Let ρ : G → Aut E[ℓ] be the Galois representation on the ℓ torsion of E. Since E is semistable for p ≠ ℓ,

ρ|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}).

Assume ρ is reducible. Then ρ has an invariant subspace so

ρ ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

Then from the form of ρ|I_{p} we see that the characters α and β could be ramified only at ℓ. Thus α = χ^{i} and β = χ^{1−i} where χ is the mod ℓ cyclotomic character. The exponents are i and 1 − i since αβ is the determinant which is χ. [[Why is χ supposed to be the only possible unramified character? probably since whatever the character is, it is a product of χ times something else, and the other factor is ramified.]]

What happens to ρ at ℓ, i.e., what is ρ|I_{ℓ}? There are only two possibilities. Either ρ|I_{ℓ} is the direct sum of the two fundamental characters or it is the sum of the trivial character with χ. The second possibility must be the one which occurs. [[I do not understand why... something about “characters globally are determined by local information.”]] So either i = 0 or i = 1. If i = 0,

ρ = (\begin{matrix}1 & ∗ \\ 0 & χ \end{matrix}).

This means that there is an element of E[ℓ] whose subspace is left invariant under the action of Galois. Thus E has a point of order ℓ rational over Q. If i = 1 then

ρ = (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix}).

Therefore μ_{ℓ} ↪ E[ℓ]. Divide to obtain E′ = E/μ_{ℓ}. The representation on E[ℓ]/μ_{ℓ} is constant (this is basic linear algebra) so the resulting representation on E′[ℓ] has the form ( \begin{matrix}1 & ∗ \\ 0 & χ \end{matrix} ) so E′ has a rational point of order ℓ.

Now E[2] is a trivial Galois module since it contains 3 obvious rational points, namely (0, 0), (A, 0), and (B, 0). Thus the group structure on the curve E (or E′) (which we constructed from a counterexample to Fermat) contains

Z/2Z ⊕ Z/2Z ⊕ Z/ℓZ.

In Mazur’s paper [[“rational isogenies of prime degree”]] he proves that ℓ ≤ 3. Since we assumed that ℓ > 5, this is a contradiction. Notice that we have not just proved FLT. We have demonstrated the irreducibility of the Galois representation on the ℓ torsion of the elliptic curve E arising from a hypothetical counterexample to FLT.

We now have a representation

ρ : Gal(\overline{Q}/Q) → GL(2, F_{ℓ}) = Aut E[ℓ]

which is irreducible and modular of weight 2 and level N = N_{E} (the conductor of E). Because ρ is irreducible we conclude that ρ is modular of weight k(ρ) and level N(ρ). Furthermore

ord_{ℓ} Δ_{E} = ord_{ℓ}(ABC)^{2} = 2ℓ ord_{ℓ} abc ≡ 0 (mod ℓ)

so k(ρ) = 2.

We can also prove that N(ρ) = 2. Clearly N(ρ)|N_{E}. This is because N(ρ) computed locally at p ≠ ℓ divides the power of p in the conductor of the ℓ-adic representation for E at p. [[I do not understand this.]] Since ρ is only ramified at 2 or maybe ℓ, N(ρ) must be a power of 2. For p ≠ ℓ, ρ is ramified at p if and only if ord_{p} Δ_{E} ≢ 0 (mod ℓ) which does happen when p = 2. Since N_{E} is square free this implies that N(ρ) = 2. But S_{2}(Γ_{1}(2)) = 0, which is the ultimate contradiction!

But how do we know semistable elliptic curves over Q are modular?

22.2 Modular elliptic curves

Theorem 22.2.1 (Theorem A). Every semistable elliptic curve over Q is modular.

There are several ways to define a modular elliptic curve.

Definition 22.2.2. Let E be an elliptic curve. Then E is modular if there is a prime ℓ > 2 such that the associated ℓ-adic Galois representation

ρ_{E,ℓ^{∞}} : G = Gal(\overline{Q}/Q) → GL(2, Z_{ℓ})

defined by the ℓ-power division points on E is modular (i.e., it is a ρ_{f,λ}).

Definition 22.2.3. Let E be an elliptic curve of conductor N_{E}. For each prime p not dividing N_{E} one associates a number a_{p} related [[in a simple way!]] to the number of points on the reduction of E modulo p. Then E is modular if there exists a cusp form f = ∑ b_{n}q^{n} which is an eigenform for the Hecke operators such that a_{p} = b_{p} for almost all p. In the end one deduces that f can be chosen to have weight 2, trivial character, and level N_{E}.

Definition 22.2.4 (Shimura). An elliptic curve E is modular if there is nonconstant map X_{0}(N) → E for some N.

Theorem 22.2.5 (Theorem B. Wiles, Taylor-Wiles). Suppose E is a semistable elliptic curve over Q and suppose ℓ is an odd prime such that E[ℓ] is irreducible and modular. Then ρ_{E,ℓ^{∞}} is modular and hence E is modular.

Now we sketch a proof that theorem B implies theorem A. First take ℓ = 3. If E[3] is irreducible then by work of Langlands-Tunnel we win. The idea is to take

ρ : G → GL(2, F_{3}) ↪ GL(2, Z[\sqrt{−2}]) ⊂ GL(2, C).

The point is that there are two maps Z[\sqrt{−2}] → F_{3} given by reduction modulo each of the primes lying over 3. Choosing one gives a map GL(2, Z[\sqrt{−2}]) → GL(2, F_{3}) which, for some amazing reason [[which is?]], has a section. Then ρ : G → GL(2, C) is a continuous representation with odd determinant that must still be irreducible. By Langlands-Tunnel we know that ρ is modular and in fact ρ comes from a weight 1 cusp form f of level a power of 3 times powers of most primes dividing cond(E). Reducing f modulo some prime of Z[\sqrt{−2}] lying over 3 we obtain a mod 3 modular form which corresponds to ρ : G → E[3]. The proof of all this uses the immense base-change business in Langlands’ book. [[Ribet next says: “have to get 3’s out of the level! This jacks up the weight, and the level is still not square free. Then have to adjust the weight again.” I do not know what the point of this is.]]

Kevin Buzzard asked a question relating to how one knows the hypothesis needed for the theorem on weights and levels applies in our situation. To answer this, suppose ℓ = 3 or 5. Form the associated representation ρ : G → GL(2, F_{ℓ}) coming from E[ℓ] and assume it is irreducible, modular and semistable.

Definition 22.2.6. A mod ℓ Galois representation is semistable if for all p ≠ ℓ, the inertia group I_{p} acts unipotently and the conjectured weight is 2 or ℓ + 1.

Note that det ρ = χ.

Lemma 22.2.7. Under the above assumptions, ℓ divides the order of the image of ρ.

Proof. If not, then ρ is finite at all primes p, since for primes p ≠ ℓ inertia acts trivially [[some other argument for ℓ]]. Inertia acts trivially because if the order of the image of ρ is prime to ℓ then ρ acts diagonally. For if not then since ρ|I_{p} is unipotent (hence all eigenvalues are 1), in a suitable basis something like ( \begin{matrix}1 & ψ \\ 0 & 1 \end{matrix} ) is in the image of ρ and has order ℓ, a contradiction. Because of finiteness k(ρ) = 2 and N(ρ) = 1 which is a contradiction since there are no forms of weight 2 and level 1.

Next we consider what happens if E[3] is reducible. There are two cases to consider. First suppose E[5] is also reducible. Then E contains a rational subgroup of order 15. We can check by hand that all such curves are modular. The key result is that X_{0}(15)(Q) is finite.

The second possibility is that E[5] is irreducible. In this case we first find a curve E′ which is semistable over Q such that

• E′[5] ≅ E[5] (this is easy to do because of the lucky coincidence that X_{0}(5) has genus 0)

• E′[3] is irreducible

Next we discover that E′ is modular since E′[3] is irreducible. This implies E′[5] is modular hence E[5] is modular. Theorem B then implies E is modular.

Deformations

23.1 Introduction

For the rest of the semester let ℓ be an odd prime. Let ρ : G → GL(2, F_{ℓ^{ν}}) be such that

• ρ is modular

• ρ is irreducible

• ρ is semistable Definition 23.1.1. The representation ρ is semistable if

• for all p ≠ ℓ,

ρ|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}),

(∗ is typically trivial since most primes are unramified.)

• k(ρ) = 2 or ℓ + 1. This means that there are 2 possibilities.

1. ρ is finite at D_{ℓ}.

2.

ρ|D_{ℓ} ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

where β is unramified. (Since det(ρ) = χ we can add that α|I_{ℓ} = χ.) If k(ρ) = 2 then possibility 1 occurs. If k(ρ) = ℓ + 1 then we are in case 2, but being in case 2 does not imply that k(ρ) = ℓ + 1. If ρ comes from an elliptic curve E/Q, then case 1 occurs if E has good reduction at ℓ whereas case 2 occurs if E has ordinary multiplicative reduction [[I am not sure about this last assertion because I missed it in class.]].

What is a deformation of ρ and when can we prove that it is modular?

Let A be a complete local Noetherian ring with maximal ideal 𝔪 and residue field F_{ℓ^{ν}} (so A is furnished with a map A/𝔪 ⟶^{∼} F_{ℓ^{ν}}). Let

ρ̃_{A} : G → GL(2, A)

be a representation which is ramified at only finitely many primes. Assume ρ̃ = ρ̃_{A} lifts ρ, i.e., the reduction of ρ̃ mod 𝔪 gives ρ.

Theorem 23.1.2. Let the notation be as above. Then ρ̃ is modular if and only if it satisfies (∗).

Neither of the terms in this theorem have been defined yet.

23.2 Condition (∗)

Let

ρ : G = Gal(\overline{Q}/Q) → GL(2, F_{ℓ^{ν}})

be a Galois representation.

The statement we wish to understand is

“All ρ̃ which satisfy (∗) are modular.”

Let A be a complete local Noetherian ring with residue field F_{ℓ^{ν}}. This means that we are given a map A/𝔪 ≅ F_{ℓ^{ν}}. Suppose ρ̃ : G → GL(2, A) satisfies the following conditions:

• ρ̃ lifts ρ,

• det ρ̃ = χ̃,

• ρ̃ is ramified at only finitely many primes, and

• condition (∗).

What is condition (∗)? It the requirement that ρ̃ have the same qualitative properties as ρ locally at ℓ. There are two cases to consider.

Case 1. (arising from supersingular reduction at ℓ) Suppose ρ is finite and flat at ℓ. Then ρ|I_{ℓ} is given by the 2 fundamental characters

I_{ℓ} → F_{ℓ^{2}}^{∗}

of level 2 (instead of from powers of these characters because of the semistability assumption). Condition (∗) is that the lift of ρ is also constrained to be finite and flat. This means that for every n ≥ 1,

ρ̃|D_{ℓ} mod 𝔪^{n} : D_{ℓ} → GL(2, A/𝔪^{n})

is finite and flat, i.e., it comes from a finite flat group scheme over Z_{ℓ} which is provided with an action of A/𝔪^{n} as endomorphisms.

Case 2. (bad multiplicative reduction at ℓ or good ordinary reduction at ℓ) In case 2

ρ|D_{ℓ} ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

where β is unramified (equivalently α|I_{ℓ} = χ). [[In the case of good ordinary reduction β is given by the action of Galois on E[ℓ] in characteristic ℓ.]] Condition (∗) is the requirement that

ρ̃|D_{ℓ} ≅ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix}),

where β̃ is unramified, α̃|I_{ℓ} = χ̃. It follows automatically that β̃ lifts β.

23.2.1 Finite flat representations

In general what does it mean for ρ to be finite and flat? It means that there exists a finite flat group scheme G over Z_{ℓ} such that G(\overline{Q}_{ℓ}) is the representation space of ρ. This definition is subtle.

Coleman asked if there is a way to reformulate the definition without mentioning group schemes. Ribet mentioned Hopf algebras but then stopped. Buzzard suggested some of the subtlety of the definition by claiming that in some situation χ is finite flat whereas χ^{2} is not. Ogus vaguely conjectured that Fontaine’s language is the way to understand this.

It is possible in case 2 above for ρ|D_{ℓ} to be finite flat without ρ̃ finite flat. The quintessential example is an elliptic curve E with supersingular reduction at ℓ such that ℓ| ord_{ℓ} Δ_{E}.

23.3 Classes of liftings

Let Σ be a finite set of prime numbers. We characterize a class of liftings ρ̃ which depends on Σ. What does it mean for ρ̃ to be in the class of deformations corresponding to Σ?

23.3.1 The case p ≠ ℓ

First we talk about the case when p ≠ ℓ. If p ∈ Σ then there is no special condition on ρ̃|I_{p}. If p ∉ Σ one requires that ρ̃ is qualitatively the same as ρ. This means

1. If ρ is unramified at p (which is the usual case), then we just require that ρ̃ is unramified at p.

2. If ρ is ramified but unipotent at p so

ρ|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix})

we require that

ρ̃|I_{p} ≅ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}).

This situation can occur with an elliptic curve which has multiplicative reduction at p and for which ℓ ∤ ord_{p} Δ_{E}.

At this point I mention the prime example. Example 23.3.1. Suppose ρ̃ = ρ_{f,λ} is the λ-adic representation attached to f. What can we say about ord_{p} N(f)? We know that

ord_{p} N(f) = ord_{p} N(ρ) + dim(ρ)^{I_{p}} − dim(ρ̃)^{I_{p}}

where (ρ)^{I_{p}} means the inertia invariants in the representation space of ρ. If ρ is semistable then

ord_{p} N(ρ) + dim(ρ)^{I_{p}} = 2.

Since we are assuming ρ is semistable, ord_{p} N(f) ≤ 2. Furthermore, the condition p ∉ Σ is a way of saying

ord_{p} N(f) = ord_{p} N(ρ).

Thus the requirement that p ∉ Σ is that the error term dim(ρ)^{I_{p}} −dim(ρ̃)^{I_{p}} vanish.

Note that ord_{p} N(ρ̃) = ord_{p} N(f) by Carayol’s theorem. Thus ord_{p} N(f) is just a different way to write ord_{p} N(ρ̃). Example 23.3.2. Imagine ρ is ramified at p and ord_{p} N(ρ) = 1. Then ord_{p} N(f) is either 1 or 2. The requirement that p ∉ Σ is that ord_{p} N(f) = 1.

23.3.2 The case p = ℓ

Next we talk about the case p = ℓ. There are two possibilities: either ℓ ∈ Σ or ℓ ∉ Σ. If ℓ ∈ Σ then we impose no further condition on ρ̃ (besides the already imposed condition (∗), semistability at ℓ, etc.). If ℓ ∉ Σ and ρ if finite and flat (which is not always the case) then we require ρ̃ to be finite and flat. If ℓ ∉ Σ and ρ is not finite flat then no further restriction (this is the Tate curve situation).

Suppose ρ̃ = ρ_{f,λ}, and ρ̃ belongs to the class defined by Σ. We want to guess (since there is no Carayol theorem) an integer N_{Σ} such that N(f)|N_{Σ}. What is N_{Σ}? It will be

N_{Σ} = ∏_{\begin{matrix}p≠ℓ \\ p∈Σ \end{matrix}} p^{2} · ∏_{\begin{matrix}p≠ℓ \\ p∉Σ \end{matrix}} p^{ord_{p} N(ρ)} · ℓ^{δ}.

Here ord_{p} N(ρ) is 1 if and only if ρ is ramified at p. The exponent δ is either 0 or 1. It is 1 if and only if k(ρ) = ℓ + 1 or ℓ ∈ Σ and ρ is ordinary at ℓ, i.e.,

ρ|D_{ℓ} ≅ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix}).

[[This definition could be completely wrong. It was definitely not presented clearly in class.]]

There is an exercise associated with this. It is to justify a priori the definition of δ. Suppose, for example, that ρ_{f,λ} satisfies (∗), then we want to show that ℓ^{2} ∤ N(f).

Theorem 23.3.3. Every ρ̃ of class Σ comes from S_{2}(Γ_{0}(N_{Σ})).

Define an approximation T to the Hecke algebra by letting

T = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N_{Σ})))

where we adjoin only those T_{n} for which (n, ℓN_{Σ}) = 1. For some reason there exists a map

T → F_{ℓ^{ν}} : T_{r} ↦ tr ρ(Frob_{r}).

Why should such a map exist? The point is that we know by the theorem that ρ comes by reduction from a ρ_{f,λ} with f ∈ S_{2}(Γ_{0}(N_{Σ})).

But there is a wrinkle. [[I do not understand: He says: “Clearly N(ρ)|N_{Σ}. k(ρ) = 2 or ℓ + 1. If k(ρ) = ℓ + 1 then δ = 1 so ℓ|N_{Σ}. Have to slip over to weight 2 in order to get f.” This does not make any sense to me.]]

23.4 Wiles’s Hecke algebra

Composing the map T → O_{E_{f}} with reduction mod λ from O_{E_{f}} to F_{ℓ^{ν}} we obtain a map T → F_{ℓ^{ν}}. Let 𝔪 be the kernel. Then 𝔪 is a maximal ideal of T. Wiles’s Hecke algebra is the completion T_{𝔪} of T at 𝔪. [[For some reason]] there exists

ρ̃: G = Gal(\overline{Q}/Q) → GL(2, T_{𝔪})

such that tr(ρ̃(Frob_{r})) = T_{r}. What makes this useful is that ρ̃ is universal for lifts of type Σ. This means that given any lift τ of type Σ there exists a map

φ : GL(2, T_{𝔪}) → GL(2, A)

such that τ = φρ̃.

Another key idea is that the approximation T obtained by just adjoining those T_{n} with (n, ℓN_{Σ}) = 1 is, after completing at certain primes, actually equal to the whole Hecke algebra.

236 23. Deformations

The Hecke Algebra T_{Σ}

24.1 The Hecke algebra

Throughout this lecture ℓ ≠ p and ℓ ≥ 3. We are studying the representation ρ : G → GL(2, F_{ℓ^{ν}}). This is an irreducible representation, ℓ is odd, ρ is semistable, and det ρ = χ. To single out certain classes of liftings we let Σ be a finite set of primes. Let A be a complete local Noetherian ring with residue filed F_{ℓ^{ν}}. We take liftings ρ̃: G → GL(2, A) such that ρ̃ reduces down to ρ, det ρ̃ = χ̃, and ρ̃ is “like” ρ away from Σ. For example, if ρ is unramified at p we also want ρ̃ unramified at p, etc.

Assume ρ̃ is modular. We guess the serious divisibility condition that N(f)|N_{Σ}. Recall that

N_{Σ} = ∏_{\begin{matrix}p≠ℓ \\ p∈Σ \end{matrix}} p^{2} · ∏_{\begin{matrix}p≠ℓ \\ p∉Σ \end{matrix}} p^{ord_{p} N(ρ)} · ℓ^{δ}.

To define δ consider two cases.

• level 1 case. Take δ = 1 if ℓ ∈ Σ or if ρ is not finite at ℓ. Take δ = 0 otherwise.

• level 2 case. This is the case when ρ|I_{ℓ} has order ℓ^{2} − 1. Take δ = 0.

A priori nobody seems to know how to prove that N(f)|N_{Σ} given only that ρ̃ is modular. In the end we will show that all modular ρ̃ in fact come from

S_{2}(Γ_{0}(N_{Σ})).

This can be regarded as a proof that N(ρ)|N_{Σ}.

Last time we tried to get things going by defining the anemic Hecke algebra

T = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N_{Σ})))

238 24. The Hecke Algebra T_{Σ}

where we only adjoin those T_{n} for which (n, ℓN_{Σ}) = 1. By some level lowering theorem there exists an f ∈ S_{w}(Γ_{0}(N_{Σ})) giving ρ̃ so we obtain a map T → F. The map sends T_{n} to the reduction modulo λ of its eigenvalue on f. (λ is a prime of the ring of integers of E_{f} lying over ℓ.) [[... something about needing an f of the right level = N(ρ).]]

Let 𝔪 ⊂ T be the kernel of the above defined map T → F. Then 𝔪 is a maximal ideal. Let T_{𝔪} be the completion of T at 𝔪 and note that

T_{𝔪} ↪ T ⊗_{Z} Z_{ℓ}.

We need to know that T_{𝔪} is Gorenstein. This is done by comparing T_{𝔪} to some full Hecke ring. Thus

T ⊂ R = Z[…, T_{n}, …] ⊂ End(S_{2}(Γ_{0}(N_{Σ})))

where the full Hecke ring R is obtained by adjoining the T_{n} for all integers n. We should think of R as

R = T[T_{ℓ}, {U_{P} : p|N_{Σ}}].

Note that T_{ℓ} may or may not be a U_{ℓ} depending on if ℓ|N_{Σ}.

Lemma 24.1.1. If ℓ ∤ N_{Σ} then R = T[U_{p}, …]. Thus if T_{ℓ} is not a U_{ℓ} then we do not need T_{ℓ}.

Proof. “This lemma is an interesting thing and the proof goes as follows. Oooh. Sorry, this is not true. Ummm.... hmm.”

The ring T[U_{p}, …] is clearly of finite index in R since: if q is a random prime number consider R ⊗_{Z} Q_{q} compared to T[U_{p}, …] ⊗_{Z} Q_{q}. [[I do not know how to do this argument. The lemma as stated above probably isn’t really true. The point is that the following lemma is the one we need and it is true.]]

Let T[U_{p}, …] be the ring obtained by adjoining to T just those U_{p} with p|N_{Σ}.

Lemma 24.1.2. If ℓ ∤ N_{Σ} then the index (R : T[U_{p}, …]) is prime to ℓ. Note that we assume ℓ ≥ 3.

Proof. We must show that the map T[U_{p}, …] → R/ℓR is surjective. [[I thought about it for a minute and did not see why this suffices. Am I being stupid?]] Let A = F_{ℓ}[T_{n} : (n, ℓ) = 1] be the image of T[U_{p}, …] in R/ℓR so we have a diagram

\xymatrix{T[U_{p}, …] \ar[rr] \ar[dr] &  & R/ℓR \\  & A \ar[ur]}

We must show that A = R/ℓR. There is a beautiful duality

R/ℓR × S_{2}(Γ_{0}(N_{Σ}); F_{ℓ}) ⟶ F_{ℓ} (perfect pairing).

Thus A^{⊥} = 0 if and only if A = R/ℓR.

Suppose 0 ≠ f ∈ A^{⊥}, then a_{n}(f) = 0 for all n such that (n, ℓ) = 1. Thus f = ∑ a_{nℓ}q^{nℓ}. Let θ = q \frac{d}{dq} be the theta operator. Since the characteristic is ℓ, θ(f) = 0. On the other hand w(f) = 2 and since ℓ ≥ 3, ℓ ∤ 2. Thus w(θf) = w(f)+ℓ+1 = 2+ℓ+1 which is a contradiction since θf = 0 and w(0) = 0 ≠ 3+ℓ. [[The weight is an integer not a number mod ℓ, right?]]

24.2 The Maximal ideal in R 239

Example 24.1.3. The lemma only applies if ℓ ≥ 3. Suppose ℓ = 2 and consider S_{2}(Γ_{0}(23)). Then

T[U_{p}, …] = Z[\sqrt{5}] ⊂ R = Z[\frac{1 + \sqrt{5}}{2}]

so (R : T[U_{p}, …]) is not prime to 2. Remark 24.1.4. If N = N_{Σ} then

rank_{Z} T = ∑_{M|N} S_{2}(Γ_{0}(M))^{new}

and

rank_{Z} R = dim S_{2}(Γ_{0}(N)).

There is an injection

⨁_{M|N} S_{2}(Γ_{0}(M))^{new} ↪ S_{2}(Γ_{0}(N))

but ⊕S_{2}(Γ_{0}(M))^{new} is typically much smaller than S_{2}(Γ_{0}(N)).

24.2 The Maximal ideal in R

The plan is to find a special maximal ideal 𝔪_{R} of R lying over 𝔪.

𝔪_{R}  R 𝔪  T

Once we finally find the correct 𝔪 and 𝔪_{R} we will be able to show that the map T_{𝔪} → R_{𝔪_{R}} is an isomorphism. In finding 𝔪_{R} we will not invoke some abstract going up theorem but we will “produce” 𝔪_{R} by some other process. The ideal 𝔪 was defined by a newform f level M|N_{Σ}. The coefficients of f lie in

O_{λ} = (O_{E_{f}})_{λ}

where E_{f} is the coefficient ring of f and λ is a prime lying over ℓ. Thus O_{λ} is an ℓ-adic integer ring. Composing the residue class map O_{λ} → \overline{F} with the eigenvalue map T → O_{λ} we obtain the map T → \overline{F}.

In order to obtain the correct 𝔪_{R} we will make a sequence of changes to f to make some good newform. [[Is the motivation for all this that the lemma will not apply if ℓ|N_{Σ}?]]

24.2.1 Strip away certain Euler factors

Write f = ∑ a_{n}q^{n}. Replace f by

h = ∑_{certain n} a_{n}q^{n}

where the sum is over those n which are prime to each p ∈ Σ. What does this mean? If we think about the L-function L(f, s) = ∏_{p} L_{p}(f, s), then h has L-function

L(h, s) = ∏_{p∉Σ} L_{p}(f, s).

Furthermore making this change does not take us out of S_{2}(Γ_{0}(N_{Σ})), i.e., h ∈ S_{2}(Γ_{0}(N_{Σ})). [[He explained why but my notes are very incomplete. They say: Why. Because h = (f ⊗ ε) ⊗ ε, (ε Dirichlet character ramified at primes in N_{Σ}). Get a form of level lcm(∏_{p∈Σ} p^{2}, N(f))|N_{Σ}. Can strip and stay in space since N_{Σ} has correct squares built into it.]] Remark 24.2.1. Suppose that p||N_{Σ}. Then p ∉ Σ and p||N(ρ). The level of f is

N(f) = \begin{cases}N(ρ), & if k(ρ) = 2 \\ N(ρ)ℓ, & if k(ρ) = ℓ + 1 \end{cases} .

If p||N(ρ) then f|T_{p} = a_{p}(f)f. Thus h is already an eigenform for T_{p} unless p ∈ Σ in which the eigenvalue is 0.

[[I do not understand this remark. Why would the eigenvalue being 0 mean that T_{p} is not an eigenform? Furthermore, what is the point of this remark in the wider context of transforming f into a good newform.]]

24.2.2 Make into an eigenform for U_{ℓ}

We perform this operation to f to obtain a form g then apply the above operation to get the ultimate h having the desired properties. Do this if ℓ|N_{Σ} but ℓ ∤ N(f). This happens precisely if ℓ ∈ Σ and ρ is good and ordinary at ℓ. Then f|U_{ℓ} is just some random junk. Consider g = f + ∗f(q^{ℓ}) where ∗ is some coefficient. We see that if ∗ is chosen correctly then g|U_{ℓ} = Cg for some constant C. There are 2 possible choices for ∗ which lead to 2 choices for C. Let a_{ℓ} = a_{ℓ}(f), then C can be either root of

X^{2} + a_{ℓ}X + ℓ = 0.

This equation has exactly one unit root in O_{λ}. The reason is because we are in the ordinary situation so a_{ℓ} is a unit. But ℓ is not a unit. The sum of the roots is a unit but the product is not. [[Even this is not clear to me right now. Definitely check this later.]] Make the choice of ∗ so that real root is C. Then we obtain a g such that

g|U_{ℓ} = (unit) · g.

Next apply the above procedure to strip g and end up with an h such that

• h|U_{ℓ} = (unit) · h,

• h|U_{p} = 0 for p ∈ Σ (p ≠ ℓ), and

• h|T_{p} = a_{p}(f) · h for p ∉ Σ.

Now take the form h. It gives a map R → O_{λ} which extends T → O_{λ}. Let 𝔪_{R} be the kernel of the map R → \overline{F} obtained by composing R → O_{λ} with O_{λ} → \overline{F}. A lot of further analysis shows that T_{𝔪} → R_{𝔪_{r}} is an isomorphism. We end up having to show separately that the map is injective and surjective.

[[In this whole lecture p ≠ ℓ.]]

[[Wiles’s notation: His T_{𝔪_{R}} is my R and his T′ is my T_{𝔪}.

24.3 The Galois representation

We started with a representation ρ, chose a finite set of primes Σ and then made the completed Hecke algebra T_{𝔪}. Our goal is to construct the universal deformation of ρ of type Σ. The universal deformation is a representation

ρ̃: G → GL(2, T_{𝔪})

such that for all primes p with p ∤ ℓN_{Σ}, ρ̃(Frob_{p}) has trace T_{p} ∈ T_{𝔪} and determinant p.

We now proceed with the construction of ρ̃. Let

T = Z[…, T_{n}, …], (n, ℓN_{Σ}) = 1

be the anemic Hecke algebra. Then T ⊗ Q decomposes as a product of fields

T ⊗ Q = ∏_{f} E_{f}

where the product is over a set of representatives for the Galois conjugacy classes of newforms of weight 2, trivial character, and level dividing N_{Σ}. Since T is integral (it is for example a finite rank Z-module), T ↪ ∏_{f} O_{f}. Since Z_{ℓ} is a flat Z-module,

T ⊗ Z_{ℓ} ↪ ∏_{f} O_{f} ⊗ Z_{ℓ} = ∏_{f,λ} O_{f,λ}

where the product is over a set of representatives f and all λ|ℓ.

T_{𝔪} is a direct factor of T ⊗ Z_{ℓ}. [[This is definitely not the assertion that T_{𝔪} is an O_{f,λ}. What exactly is it the assertion of really?]]

We can restrict the product to a certain finite set S and still obtain an injection

T_{𝔪} ↪ ∏_{(f,λ)∈S} O_{f,λ}.

The finite set S consists of those (f, λ) such that the prime λ of O_{f} pulls back to 𝔪 under the map T → O_{f} obtained by composing T → ∏_{f} O_{f} with the projection onto O_{f}. [[Why is this enough so that T_{𝔪} still injects in?]] Restricting to a finite product is needed so that

[ ∏_{(f,λ)∈S} O_{f,λ} : T_{𝔪}] < ∞.

Given f and λ there exists a representation

ρ_{f,λ} : G → GL(2, O_{f,λ}).

It is such that tr ρ_{f,λ}(Frob_{p}) = a_{p} is the image of T_{p} under the inclusion

T ⊗ Z_{ℓ} ↪ ∏_{(f,λ)∈S} O_{f,λ}.

Put some of these ρ_{f,λ} together to create a new representation

∏_{(f,λ)∈S} ρ_{f,λ} : G ⟶^{ρ′} GL(2, ∏ O_{f,λ}) ⊂ GL(2, T_{𝔪} ⊗ Q).

The sought after universal deformation ρ̃ is a map making the following diagram commute

\xymatrix{G \ar[rr]^{ρ′} \ar[dr]_{ρ̃} &  & GL(2, T_{𝔪} ⊗ Q) \\  & GL(2, T_{𝔪}) \ar@{^{(}->}[ur]}

Theorem 24.3.1. ρ′ is equivalent to a representation taking values in GL(2, T_{𝔪}).

One way to [[try to]] prove this theorem is by invoking a general theorem of Carayol. [[and then what? does this way work? why is it not a good way?]] But the right way to prove the theorem is Wiles’s way.

24.3.1 The Structure of T_{𝔪}

Just as an aside let us review the structure of T_{𝔪}.

• T_{𝔪} is local.

• T_{𝔪} is not necessarily a discrete valuation ring.

• T_{𝔪} ⊗ Q is a product of finite extensions of Q_{ℓ}.

• T_{𝔪} is not necessarily a product of rings O_{f,λ}.

• T_{𝔪} need not be integral.

24.3.2 The Philosophy in this picture

Choose c to be a complex conjugation in G = Gal(\overline{Q}/Q). Since ℓ is odd det(c) = −1 is a very strong condition which rigidifies the situation.

24.3.3 Massage ρ

Choose two 1-dimensional subspaces so that

ρ(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})

with respect to any basis consisting of one vector from each subspace. For any σ ∈ G write

ρ(σ) = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix}).

Then a_{σ}d_{σ} and b_{σ}c_{σ} are somehow intrinsically defined. This is because

ρ(σc) = (\begin{matrix}−a_{σ} & ? \\ ? & d_{σ} \end{matrix})

so

a_{σ} = \frac{tr(ρ(σ)) − tr(ρ(σc))}{2}

and

d_{σ} = \frac{tr(ρ(σ)) + tr(ρ(σc))}{2}.

Since we know the determinant it follows that b_{σ}c_{σ} is also intrinsically known. [[The point is that we know certain things about these matrices in terms of their traces and determinants.]]

Proposition 24.3.2. There exists g ∈ G such that b_{g}c_{g} ≠ 0.

Proof. Since ρ is irreducible there exists σ_{1} such that b_{σ_{1}} ≠ 0 and there exists σ_{2} such that c_{σ_{2}} ≠ 0. If b_{σ_{2}} ≠ 0 or c_{σ_{1}} ≠ 0 then we are done. So the only problem case is when b_{σ_{1}} = 0 and c_{σ_{2}} = 0. Easy linear algebra shows that in this situation g = σ_{1}σ_{2} has the required property.

Now rigidify by choosing a basis so that b_{g} = 1. Doing this does not fix a basis because they are many ways to choose such a basis.

24.3.4 Massage ρ′

Choose a basis of (T_{𝔪} ⊗ Q)^{2} so that

ρ′(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}).

For any σ ∈ G write

ρ′(σ) = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix})

Using an argument as above shows that a_{σ}, d_{σ} ∈ T_{𝔪} since the traces live in T_{𝔪}. Furthermore b_{σ}c_{σ} ∈ T_{𝔪} since the determinant is in T_{𝔪}.

The key observation is that b_{σ}c_{σ} reduces mod 𝔪 to give the previous b_{σ}c_{σ} ∈ F corresponding to ρ(σ). This is because the determinants and traces of ρ′ are lifts of the ones from ρ. Since 𝔪 is the maximal ideal of a local ring and b_{g}c_{g} reduces mod 𝔪 to something nonzero it follows that b_{g}c_{g} is a unit in T_{𝔪}.

Choose a basis so that

ρ′(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})

and also so that

ρ′(g) = (\begin{matrix}a_{g} & 1 \\ u & d_{g} \end{matrix}) ∈ GL(2, T_{𝔪}).

Here u is a unit in T_{𝔪}.

Proposition 24.3.3. Write

ρ′(σ) = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix})

with respect to the basis chosen above. Then a_{σ}, b_{σ}, c_{σ}, d_{σ} ∈ T_{𝔪}.

Proof. We already know that a_{σ}, d_{σ} ∈ T_{𝔪}. The question is how to show that b_{σ}, c_{σ} ∈ T_{𝔪}. Since

ρ′(σg) = (\begin{matrix}a_{σ}a_{g} + b_{σ}u & ? \\ ? & c_{σ} + d_{σ}d_{g} \end{matrix})

we see that a_{σ}a_{g} + b_{σ}u ∈ T_{𝔪}. Since a_{σ}a_{g} ∈ T_{𝔪} it follows that b_{σ}u ∈ T_{𝔪}. Since u is a unit in T_{𝔪} this implies b_{σ} ∈ T_{𝔪}. Similarly c_{σ} + d_{σ}d_{g} ∈ T_{𝔪} so c_{σ} ∈ T_{𝔪}.

As you now see, in this situation we can prove Carayol’s theorem with just some matrix computations. [[This is basically a field lowering representation theorem. The thing that makes it easy is that there exists something (namely c) with distinct eigenvalues which is rational over the residue field. Schur’s paper, models over smaller fields. “Schur’s method”.]]

24.3.5 Representations from modular forms mod ℓ

If you remember back in the 70’s people would take an f ∈ S_{2}(Γ_{0}(N); F) which is an eigenform for almost all the Hecke operators

T_{p}f = c_{p}f for almost all p, and c_{p} ∈ F.

The question is then: Can you find

ρ : G → GL(2, F)

such that

tr(ρ(Frob_{p})) = c_{p} and det(ρ(Frob_{p})) = p

for all but finitely many p? The answer is yes. The idea is to find ρ by taking ρ_{f,λ} [[which was constructed by Shimura?]] for some f and reducing mod λ. The only special thing that we need is a lemma saying that the eigenvalues in characteristic ℓ lift to eigenvalues in characteristic 0.

24.3.6 Representations from modular forms mod ℓ^{n}

Serre and Deligne asked: “What happens mod ℓ^{n}?”

More precisely, let R be a local finite Artin ring such that ℓ^{n}R = 0 for some n. Take f ∈ S_{2}(Γ_{0}(N); R) satisfying the hypothesis

{r ∈ R : rf = 0} = {0}.

This is done to insure that certain eigenvalues are unique. Assume that for almost all p one has T_{p}f = c_{p}f with c_{p} ∈ R. The problem is to find ρ : G → GL(2, R) such that

tr(ρ(Frob_{p})) = c_{p} and det(ρ(Frob_{p})) = p

for almost all p.

The big stumbling block is that ρ need not be the reduction of some ρ_{g,λ} for any g, λ. [[I couldn’t understand why – I wrote “can mix up f’s from characteristic 0 so can not get one which reduces correctly.”]]

Let T = Z[…, T_{p}, …] where we only adjoin those T_{p} for which f is an eigenvector [[I made this last part up, but it seems very reasonable]]. Then f obviously gives a rise to a map

T → R : T ↦ eigenvalue of T on f .

24.4 ρ′ is of type Σ 245

The strange hypothesis on f insures that the eigenvalue is unique. Indeed, suppose T f = af and T f = bf, then 0 = T f −T f = af −bf = (a−b)f so by the hypothesis a − b = 0 so a = b.

Since the pullback of the maximal ideal of R is a maximal ideal of T we get a map T_{𝔪} → R for some 𝔪. [[I do not understand why we suddenly get this map and I do not know why the pullback of the maximal ideal is maximal.]]

Now the problem is solved. Take ρ′ : G → GL(2, T_{𝔪}) with the sought after trace and determinant properties. Then let ρ be the map obtained by composing with the map GL(2, T_{𝔪}) → GL(2, R).

24.4 ρ′ is of type Σ

Let ρ be modular irreducible and semistable mod ℓ representation with ℓ > 2. Let Σ be a finite set of primes. Then N(ρ)|N_{Σ}. We constructed the anemic Hecke algebra T which contains a certain maximal ideal 𝔪. We then consider the completion T_{𝔪} of T at 𝔪. Next we constructed

ρ′ : G → GL(2, T_{𝔪})

lifting ρ. Thus the diagram

\xymatrix{G \ar[r]^{ρ′} \ar[dr]_{ρ} & GL(2, T_{𝔪}) \ar[d] \\  & GL(2, F)}

commutes.

Some defining properties of ρ′ are

• det ρ′ = χ̃.

• tr ρ′(Frob_{r}) = T_{r}. Since topologically T_{𝔪} is generated by the Frob_{r} this is a tight condition.

• ρ′ is a lift of type Σ.

To say ρ′ is a lift of type Σ entails that ρ′ is unramified outside primes p|N_{Σ}. This is true because ρ′ is constructed from various ρ_{f,λ} with N(f)|N_{Σ}. If p ≠ ℓ and p|N(ρ) then p||N_{Σ}. Recall that

ρ|D_{p} ∼ (\begin{matrix}α & ∗ \\ 0 & β \end{matrix})

where α = βχ and α and β are unramified. For ρ′ to be a lift of type Σ we require that

ρ′|D_{p} ∼ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix})

where α̃ and β̃ are unramified lifts and α̃ = β̃χ. [[I find it mighty odd that α is χ times an unramified character and yet α is not ramified! How can that be?

Restricted to inertia β and α would be trivial but χ would not be.]] Is this true of ρ′? Yes since by a theorem of Langlands the factors

ρ_{f,λ}|D_{p} ∼ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix}) .

[[Ribet said more about this but it does not form a cohesive whole in my mind. Here is what I have got. Since ρ_{f,λ} obviously ramified at p, p||N(f)|N_{Σ}. ρ_{f,λ}|D_{p} is like an elliptic curve with bad multiplicative reduction at p. That ρ_{f,λ}|D_{p} ∼ ( \begin{matrix}a & b \\ c & d \end{matrix} ) α̃∗0β̃ really comes down to Deligne-Rapaport. If write f = ∑ a_{n}q^{n}, then a_{p} ≠ 0 and β̃(Frob_{p}) = a_{p}, α̃(Frob_{p}) = pa_{p}. Thus a_{p}^{2} = 1 since α̃β̃ = χ. Thus a_{p} = ±1 and a_{p} mod λ = β(Frob_{p}) = ±1 independent of (f, λ). So we have these numbers a_{p} = a_{p}(f) = ±1, independent of λ. ]]

24.5 Isomorphism between T_{𝔪} and R_{𝔪_{R}}

Let T ⊂ R = Z[…, T_{n}, …] be the anemic Hecke algebra with maximal ideal 𝔪. The difference between T and R is that R contains all the Hecke operators whereas T only contains the T_{p} with p ∤ ℓN_{Σ}. Wiles proved that the map T_{𝔪} → R_{𝔪_{R}} is an isomorphism. Which Hecke operators are going to hit the missing T_{p}? If we do the analysis in R_{𝔪_{R}} we see that [[I think for p ≠ ℓ!]]

T_{p} = \begin{cases}±1, & for p|N_{Σ}, p ∉ Σ \\ 0, & for p ∈ Σ \end{cases} .

This takes care of everything except T_{ℓ}. In proving the surjectivity of T_{𝔪} → R_{𝔪_{R}} we are quite happy to know that T_{p} = ±1 or 0. The nontrivial proof is given in [DDT94].

Consider the commuting diagram

\xymatrix{T_{𝔪} \ar@{^{(}->}[r] \ar[dr] & ∏ O_{f,λ} \\  & R_{𝔪_{R}} \ar[u]}

The map R_{𝔪_{R}} → ∏ O_{f,λ} is constructed by massaging f by stripping away certain Euler factors so as to obtain an eigenvector for all the Hecke operators. This diagram forces T_{𝔪} → R_{𝔪_{R}} to be injective.

[[Ogus: Is it clearly surjective on the residue field? Ribet: Yes. Ogus: OK, then we just need to prove it is ètale.]]

From the theory of the θ operator we already know two-thirds of the times that T contains T_{ℓ}.

Suppose ℓ|N_{Σ}. This entails that we are in the ordinary case, ρ is not finite at ℓ, or ℓ ∈ Σ. We did not prove in this situation that T_{ℓ} ∈ Z[…, T_{n}, …: (n, ℓ) = 1].

Using generators and relations and brute force one shows that R_{𝔪_{R}} → ∏ O_{f,λ} is an injection. Then we can compare everything in ∏ O_{f,λ}. Now

ρ_{f,λ}|D_{ℓ} ∼ (\begin{matrix}α̃ & ∗ \\ 0 & β̃ \end{matrix})

and

β̃(Frob_{ℓ}) = T_{ℓ} ∈ ∏ O_{f,λ}.

Using arguments like last time one shows that β̃(Frob_{ℓ}) can be expressed in terms of the traces of various operators. This proves surjectivity in this case.

Ultimately we have T_{𝔪} ≅ R_{𝔪_{R}}. The virtue of T_{𝔪} is that it is generated by traces. The virtue of R_{𝔪_{R}} is that it is Gorenstein. We have seen this if ℓ ∤ N_{Σ}. In fact it is Gorenstein even if ℓ||N_{Σ}. [[Ribet: As I stand here today I do not know how to prove this last assertion in exactly one case. Ogus: You mean there is another gap in Wiles’s proof. Ribet: No, it is just something I need to work out.]] When ℓ||N_{Σ} there are 2 cases. Either ρ is not finite at ℓ or it is. The case when ρ is not finite at ℓ was taken care of in [MR91]. A proof that R_{𝔪_{R}} is Gorenstein when ρ is finite at ℓ (ℓ ∈ Σ) is not in the literature.

Now forget R_{𝔪_{R}} and just think of T_{𝔪} in both ways: trace generated and Gorenstein.

24.6 Deformations

Fix an absolutely irreducible modular mod ℓ representation ρ and a finite set of primes Σ. Consider the category C of complete local Noetherian W (F)-algebras A (with A/𝔪 = F). Here F = F_{ℓ^{ν}} and W (F) is the ring of Witt vectors, i.e., the ring of integers of an unramified extension of Q_{ℓ} of degree ν.

Define a functor F : C → Set by sending A in C to the set of equivalence classes of lifts

ρ̃: G → GL(2, A)

of ρ of type Σ. The equivalence relation is that ρ̃_{1} ∼ ρ̃_{2} if and only if there exists

M ∈ GL(2, A) with M ≡ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) (mod 𝔪) such that ρ̃_{1} = M^{−1}ρ̃_{2}M.

Mazur proved [Maz89] that F is representable.

Theorem 24.6.1. There exists a lift

ρ_{univ} : G → GL(2, R_{Σ})

of type Σ such that given any lift ρ̃ : G → GL(2, A) there exists a unique homomorphism φ : R_{Σ} → A such that φ∘ρ_{univ} ∼ ρ̃ in the sense of the above equivalence relation.

Lenstra figured out how to concretely construct R_{Σ}.

Back in the student days of Ribet and Ogus, Schlessinger wrote a widely quoted thesis which gives conditions under which a certain class of functors can be representable. Mazur checks these conditions in his paper.

[[Buzzard: What happened to Schlessinger anyways? Ribet: He ended up at University of North Carolina, Chapel Hill.]]

We have constructed ρ̃ = ρ′ : G → GL(2, T_{𝔪}). By the theorem there exists a unique morphism φ : R_{Σ} → T_{𝔪} such that ρ′ = φ ∘ ρ_{univ}.

Theorem 24.6.2. φ is an isomorphism thus ρ′ is the universal deformation and T_{𝔪} is the universal deformation ring.

This will imply that any lift of type Σ is modular.

The morphism φ is surjective since

T_{p} = tr ρ̃(Frob_{p}) = tr φ ∘ ρ_{univ}(Frob_{p}) = φ(tr(ρ_{univ}(Frob_{p}))).

We have two very abstractly defined local Noetherian rings. How would you prove they are isomorphic? Most people would be terrified by this question. Wiles dealt with it.

24.7 Wiles’s main conjecture

“We are like a train which is trying to reach Fermat’s Last Theorem. Of course it has not made all of its scheduled stops. But it is on its way.”

We have a representation ρ : G → GL(2, F). Take F = F_{ℓ} for our applications today. Then the ring of Witt vectors is W (F) = Z_{ℓ}. The Hecke algebra can be embedded as

T_{𝔪} ⊂ ∏_{(g,μ)∈A} O_{g,μ}.

The Hecke algebra T_{𝔪} has the following properties.

• The index of T_{𝔪} in ∏ O_{g,μ} is finite.

• Gorenstein as a Z_{ℓ}-module, i.e., there exists an isomorphism Hom_{Z_{ℓ}}(T_{𝔪}, Z_{ℓ}) ≅ T_{𝔪}.

• T_{𝔪} is generated by the T_{r} with r prime and (r, ℓN_{Σ}) = 1.

We have constructed a representation

ρ′ : G → GL(2, T_{𝔪}).

Composing appropriately with the map T_{𝔪} ↪ ∏ O_{g,μ} gives a map

G → ∏_{(g,μ)} GL(2, O_{g,μ}).

This is the product of representations ∏ ρ_{g,μ}. The triangle is

\xymatrix{G \ar[r]^{ρ′} \ar[dr]_{∏ ρ_{g,μ}} & GL(2, T_{𝔪}) \ar[d] \\  & ∏ GL(2, O_{g,μ})}

Moreover, ρ′ is a deformation of ρ of type Σ so it lifts ρ and satisfies certain “nice as ρ” properties at primes p ∉ Σ.

Let

ρ_{univ} : G → GL(2, R_{Σ})

be the universal deformation of ρ of type Σ. Lenstra gave a very concrete paper [dSL97] constructing this ρ_{univ}. Before his paper there was only Schlesinger’s thesis. By the definition of ρ_{univ} there exists a unique map φ : R_{Σ} → T_{𝔪} = T_{Σ} such that φ ∘ ρ_{univ} = ρ′. By φ ∘ ρ_{univ} we mean the composition of ρ_{univ} with the map GL(2, R_{Σ}) → GL(2, T_{𝔪}) induces by φ. As noted last time it is easy to see that φ is surjective.

Theorem 24.7.1 (Wiles’s main ‘conjecture’). φ is an isomorphism (for each Σ).

The theorem implies the following useful result.

Theorem 24.7.2. Suppose E is a semistable elliptic curve over Q and that for some ℓ > 2 the representation ρ = ρ_{ℓ,E} on E[ℓ] is irreducible and modular. Then E is modular.

Proof. The representation

ρ̃ = ρ_{ℓ^{∞},E} : G → GL(2, Z_{ℓ})

on the ℓ-power torsion E[ℓ^{∞}] = ∪E[ℓ^{n}] is a lift of

ρ : G → Aut(E[ℓ]) = GL(2, F_{ℓ}).

Furthermore, ρ̃ is a deformation of ρ of type Σ. Applying universality and using the theorem that R_{Σ} ≅ T_{Σ} we get a map

T_{Σ} → Z_{ℓ} : T_{r} ↦ a_{r} = a_{r}(E) = Tr ρ̃(Frob_{r}).

The relevant diagram is

\xymatrix{R_{Σ} \ar[d] \ar[dr] \\ T_{Σ} \ar[r] & Z_{ℓ}}

where the map R_{Σ} → Z_{ℓ} is given by

Tr ρ_{univ}(Frob_{r}) ↦ a_{r}.

Now the full Hecke algebra Z[…, T_{n}, …] embeds into the completion T_{Σ}. Composing this with the map T_{Σ} → Z_{ℓ} above we obtain a map

α : Z[…, T_{n}, …] → Z_{ℓ}.

Because of the duality between the Hecke algebra and modular forms there exists a modular form h ∈ S_{2}(Γ_{0}(N_{Σ}), Z_{ℓ}) corresponding to α. Since α is a homomorphism h is a normalized eigenform. Furthermore a_{r}(h) = a_{r}(E) ∈ Z for all primes r ∤ ℓN_{Σ}. Since almost all coefficients of h are integral it follows that h is integral. Because we know a lot about eigenforms we can massage h to an eigenform in S_{2}(Γ_{0}(N_{E}), Z).

[[Some undigested comments follow.]]

• Once there is any connection between ρ and a modular form one can prove Taniyama-Shimura in as strong a form as desired. See the article Number theory as Gadfly.

• Take the abelian variety A_{h} attached to h. The λ-adic representation will have pieces with the same representations. Using Tate’s conjecture we see that E is isogenous to A_{h}. Use at some points Carayol’s theorem: If g is a form giving rising to the abelian variety A then the conductor of A is the same as the conductor of g.

• Tate proved that if two elliptic curves have isomorphic ρ_{ℓ^{∞}} for some ℓ then they are isogenous.

24.8 T_{Σ} is a complete intersection

Recall the construction of T_{Σ} = T_{𝔪}. Let T be the anemic Hecke algebra. Then

T ⊗ Z_{ℓ} ↪ ∏ O_{g,μ}

where the product is over a complete set of representatives (for the action of Galois on eigenforms) g and primes μ lying over ℓ. We found a specific (f, λ) for Σ = ∅ such that \overline{ρ}_{λ,f} = ρ. The maximal ideal 𝔪 was defined as follows. The form f induces a map T → O_{f,λ}. Taking the quotient of O_{f,λ} by its maximal ideal we obtain a map T → \overline{F}_{ℓ}. Then 𝔪 is the kernel of this map. The diagram is

\xymatrix{T \ar[d] \ar[dr] \\ O_{f,λ} \ar[r] & _{\overline{F}_{ℓ}}}

The map O_{f,λ} → \overline{F}_{ℓ} is

a_{r}(f) ↦ Tr ρ(Frob_{r}).

To fix ideas we cheat and suppose O_{f,λ} = Z_{ℓ}. [[In Wiles’s optic this is OK since he can work this way then tensor everything at the end.]]

Now ρ_{f,λ} is a distinguished deformation of ρ [[“Distinguished” is not meant in a mathematical sense]]. The map f gives rise to a map T_{Σ} → O = Z_{ℓ} which we also denote by f

\xymatrix{R_{Σ} \ar[dr] \ar[r]^{φ} & T_{Σ} \ar[d]^{f} \\  & O = Z_{ℓ}}

24.9 The Inequality #O/η ≤ #℘_{T} /℘_{T}^{2} ≤ #℘_{R}/℘_{R}^{2}

Let

ρ : G = Gal(\overline{Q}/Q) → GL(2, F_{ℓ})

be irreducible and modular with ℓ > 2. Let Σ be a finite set of primes. We assume there is a modular form f of weight 2 with coefficients in Z_{ℓ} which gives rise ρ. Let

ρ_{f,λ} : G → GL(2, Z_{ℓ})

24.9 The Inequality #O/η ≤ #℘_{T} /℘_{T}^{2} ≤ #℘_{R}/℘_{R}^{2} 251

be the representation coming from f, then ρ_{f,λ} reduces to ρ modulo ℓ.

Let R_{Σ} be the universal deformation ring, so every deformation of ρ of type Σ factors through R_{Σ} in an appropriate sense. Let T_{Σ} be the Hecke ring associated to Σ. It is a Z_{ℓ}-algebra which is free of finite rank. Furthermore

T_{Σ} ⊂ ∏_{(g,μ)∈A} O_{g,μ} = O_{f,μ} × ∏_{(g,μ)∈A−{(f,μ)}} O_{g,μ}

where A is as defined before. Define projections pr_{1} and pr_{2} onto the first and rest of the factors, respectively

pr_{1} : ∏_{(g,μ)∈A} O_{g,μ} → O = O_{f,λ}
pr_{2} : ∏_{(g,μ)∈A} O_{g,μ} → ∏_{(g,μ)≠(f,λ)} O_{g,μ}

Let φ : R_{Σ} → T_{Σ} be the map coming from the universal property of R_{Σ}. This map is surjective. The famous triangle which dominates all of the theory is

\xymatrix{R_{Σ} \ar[dr] \ar[r]^{φ} & T_{Σ} \ar[d]^{_{pr_{1}}} \\  & O = Z_{ℓ}}

24.9.1 The Definitions of the ideals

We now define two ideals. View T_{Σ} as sitting in the product ∏ O_{g,μ}.

1. The congruence ideal η ⊂ O is

η := O ∩ T_{Σ} = ker(pr_{2} : T_{Σ} → ∏_{(g,μ)≠(f,λ)} O_{g,μ})

2. The prime ideal ℘_{T} ⊂ T_{Σ} is

℘_{T} = ker(pr_{1} : T_{Σ} → O)

It is true that

#O/η ≤ #℘_{T} /℘_{T}^{2} .

The condition for equality is a theorem of Wiles.

Theorem 24.9.1. T_{Σ} is a complete intersection if and only if #O/η = #℘_{T} /℘_{T}^{2} .

There is an analogous construction for

ψ = pr_{1} ∘φ : R_{Σ} → O.

The diagram is

\begin{aligned}R_{Σ} & ⟶^{φ} & T_{Σ} \\ & ψ ↘ & ↓ pr_{1} \\ & & O \end{aligned} .

Let ℘_{R} be the kernel of ψ. From the commutativity of the above diagram we see that ψ maps ℘_{R} → → ℘_{T}. Thus we have an induced map \overline{ψ} on “tangent spaces”

\overline{ψ} : ℘_{R}/℘_{R}^{2} → → ℘_{T} /℘_{T}^{2} .

It follows that

#O/η ≤ #℘_{T} /℘_{T}^{2} ≤ #℘_{R}/℘_{R}^{2}.

There is an analogous theorem.

Theorem 24.9.2. The above inequalities are all equalities iff

• φ : R_{Σ} → T_{Σ} is an isomorphism, and

• T_{Σ} is a complete intersection ring.

24.9.2 Aside: Selmer groups

Let M be the set of matrices in M(2, Q_{ℓ}/Z_{ℓ}) which have trace 0. Then GL(2, Z_{ℓ}) operates on M by conjugation. Thus G acts on M via the representation ρ′ : G → GL(2, Z_{ℓ}). To ℘_{R}/℘_{R}^{2} there corresponds the Selmer group which is a subgroup of H^{1}(Gal(\overline{Q}/Q), M). The subgroup is

H_{Σ}^{1}(G, M) = Hom_{O}(℘_{R}/℘_{R}^{2}, Q_{ℓ}/Z_{ℓ}) ⊂ H^{1}(Gal(\overline{Q}/Q), M).

[[Since Flach’s thesis there has been a problem of trying to get an upper bound for the Selmer group. Wiles converted it into the above problem.]]

24.9.3 Outline of some proofs

We outline the key steps in the proof that #℘_{R}/℘_{R}^{2} ≤ #O/η.

Step 1: Σ = ∅

The key step is the minimal case when Σ = ∅. This is done in [TW95]. They claim to be proving the apparently weaker statement that T_{Σ} a complete intersection implies

#℘_{T} /℘_{T}^{2} = #O/η.

But in Wiles’s paper [Wil95] he obtains the inequality

#℘_{R}/℘_{R}^{2} ≤ \frac{(#℘_{T} /℘_{T}^{2} )^{2}}{#O/η} .

Combining these two shows that

#℘_{R}/℘_{R}^{2} ≤ #O/η.

In an appendix to [TW95] Faltings proves directly that

#℘_{R}/℘_{R}^{2} ≤ #O/η.

At this point there were some remarks about why Wiles might have taken a circuitous route in his Annals paper. Ribet replied,

“As Serre says, it is sometimes better to leave out any psychological behavior related to how people did something but instead just report on what they did.”

Step 2: Passage from Σ = ∅ to σ general

The second step is the induction step in which we must understand what happens as Σ grows. Thus Σ is replaced by Σ′ = Σ ∪ {q} where q is some prime not in Σ.

We will use the following notation. The object attached to Σ′ will be denoted the same way as the object attached to Σ but with a ′. Thus (℘_{R}/℘_{R}^{2})′ denotes the Selmer group for Σ′.

The change in the Selmer group when Σ is replaced by Σ′ is completely governed by some local cohomology group. There is a constant c_{q} such that

#(℘_{R}/℘_{R}^{2})′ ≤ c_{q}#(℘_{R}/℘_{R}^{2}) ≤ c_{q}#O/η.

So we just need to know that

#O/η′ ≥ c_{q}#O/η,

i.e., that η′ is small as an ideal in O. We need a formula for the ratio of the two orders.

Let T be the anemic ring of Hecke operators on S_{2}(Γ_{0}(N_{Σ})) obtained by adjoining to Z all the Hecke operators T_{n} with n prime to ℓN_{Σ}. Let T′ be the anemic Hecke ring of Hecke operators on S_{2}(Γ_{0}(N_{Σ′})).

Since N_{Σ}|N_{Σ′} there is an inclusion

S_{2}(Γ_{0}(N_{Σ})) ↪ S_{2}(Γ_{0}(N_{Σ′})).

There is one subtlety, this injection is not equivariant for all of the Hecke operators. But this is no problem because T and T′ are anemic. So the inclusion induces a restriction map r : T′ → T.

We now introduce a relative version of η which is an ideal I ⊂ T. One way to think of I is as T ∩ T′ where T and T′ are both viewed as subrings of ∏ O_{g,μ}

\xymatrix{T′ \ar[r]^{r} \ar@{^{(}->}[d] & T \ar@{^{(}->}[d] \\ T′_{𝔪′} \ar@{^{(}->}[d] & T_{𝔪} \ar@{^{(}->}[d] \\ ∏_{(g,μ)} O_{g,μ} \ar@{^{(}->}[r] & ∏_{more (g,μ)} O_{g,μ}}

The definition Lenstra would give is that

I := r(Ann_{T′}(ker(r))).

The amazing formula is

η′ = η · f(I)

where f : T → O is the map induced by the modular form f. [[After introducing this definition Ogus was very curious about how deep it is, in particular, about whether its proof uses the Gorensteiness of T. Ribet said, “somehow I do not think this formula can possibly be profound.”]]

We pause with an aside to consider Wiles’s original definition of η. By duality the map f : T_{Σ} → Z_{ℓ} induces

f^{∨} : Hom_{Z_{ℓ}}(Z_{ℓ}, Z_{ℓ}) → Hom_{Z_{ℓ}}(T_{Σ}, Z_{ℓ}) ≅ T_{Σ}.

Because T_{Σ} is Gorenstein there is an isomorphism Hom_{Z_{ℓ}}(T_{Σ}, Z_{ℓ}) ≅ T_{Σ}. Now f^{∨}(id) ∈ T_{Σ} so f(f^{∨}(id)) ∈ O = Z_{ℓ}. Wiles let η = (f(f^{∨}(id))) be the ideal generated by f(f^{∨}(id)).

To finish step 2 we must show that #O/f(I) ≥ c_{q}, i.e., that “I is small”. [[I do not see how this actually finishes step 2, but it is reasonable that it should. How does this index relate to the index of f(I)η in O?]]

Let J = J_{0}(N_{Σ}) and J′ = J_{0}(N_{Σ′}). Since

S_{2}(Γ_{0}(N_{Σ})) ↪ S_{2}(Γ_{0}(N_{Σ′}))

functoriality of the Jacobian induces a map J ↪ J′. By autoduality we also obtain an injection J^{∨} ↪ J′ and J ∩ J^{∨} is a finite subgroup of J′. [[I definitely do not understand why J is not just equal to J^{⊥}. Where does the other embedding J^{⊥} ↪ J′ come from?]]

It can be seen that J ∩ J^{∨} = J[δ] for some δ ∈ T. It turns out that

Ann_{T}(J ∩ J^{∨}) = T ∩ δ End(J) ⊇ δT.

It is an observable fact that f(δT) is an ideal of O of norm c_{q}. The heart of the whole matter is to see that the inclusion

δT ⊆ T ∩ δ End(J)

is an equality after localization at 𝔪. To do this we have to know that Tate_{𝔪}(J) ≅ T_{𝔪}^{2}. This is equivalent to the Gorenstein business. With this in hand one can just check this equality.

Unfortunately, it is the spring of 1996 and we are now 10 minutes past when the course should end. Realizing this, Ken brings the course to a close. In the grand Berkeley tradition, the room fills with applause.

Computing with Modular Forms and Abelian Varieties

256 25. Computing with Modular Forms and Abelian Varieties

The Modular Curve X_{0}(389)

Let N be a positive integer, and let X_{0}(N) be the compactified coarse moduli space that classifies pairs (E, C) where E is an elliptic curve and C is a cyclic subgroup of order N. The space X_{0}(N) has a canonical structure of algebraic curve over Q, and its properties have been very well studied during the last forty years. For example, Breuil, Conrad, Diamond, Taylor, and Wiles proved that every elliptic curve over Q is a quotient of some X_{0}(N).

The smallest N such that the Jacobian of X_{0}(N) has positive Mordell-Weil rank is 37, and Zagier studied the genus-two curve X_{0}(37) in depth in his paper [Zag85b]. From this viewpoint, the next modular curve deserving intensive investigation is X_{0}(389), which is the first modular curve whose Jacobian has Mordell-Weil rank larger than that predicted by the signs in the functional equations of the L-series attached to simple factors of its Jacobian; in fact, 389 is the smallest conductor of an elliptic curve with Mordell-Weil rank 2. Note that 389 is prime and X_{0}(389) has genus g = 32, which is much larger than the genus 2 of X_{0}(37), which makes explicit investigation more challenging.

Work of Kolyvagin [Kol88b, Kol88a] and Gross-Zagier [GZ86] has completely resolved the rank assertion of the Birch and Swinnerton-Dyer conjecture (see, e.g., [Tat66]) for elliptic curves E with ord_{s=1} L(E, s) ≤ 1. The lowest-conductor elliptic curve E that doesn’t submit to the work of Kolyvagin and Gross-Zagier is the elliptic curve E of conductor 389 mentioned in the previous paragraph. At present we don’t even have a conjectural natural construction of a finite-index subgroup of E(Q) analogous to that given by Gross and Zagier for rank 1 (but see Mazur’s work on universal norms, which might be used to construct E(Q) ⊗ Z_{p} for some auxiliary prime p).

258 26. The Modular Curve X_{0}(389)

Inspired by the above observations, and with an eye towards providing helpful data for anyone trying to generalize the work of Gross, Zagier, and Kolyvagin, in this paper we compute everything we can about the modular curve X_{0}(389). Some of the computations of this paper have already proved important in several other papers: the discriminant of the Hecke algebra attached to X_{0}(389) plays a roll in [Rib99], the verification of condition 3 in [MS01], and the remark after Theorem 1 of [?]; also, the arithmetic of J_{0}(389) provides a key example in [AS02, §4.2]. Finally, this paper serves as an entry in an “encyclopaedia, atlas or hiker’s guide to modular curves”, in the spirit of N. Elkies (see [Elk98, pg. 22]).

We hilight several surprising “firsts” that occur at level 389. The discriminant of the Hecke algebra attached to S_{2}(Γ_{0}(389)) has the apparently unusual property that it is divisible by p = 389 (see Section 26.2.1). Also N = 389 is the smallest integer such that the order of vanishing of L(J_{0}(N), s) at s = 1 is larger than predicted by the functional equations of eigenforms (see Section 26.1.3). The author conjectures that N = 389 is the smallest level such that an optimal newform factor of J_{0}(N) appears to have Shafarevich-Tate group with nontrivial odd part (see Section 26.4.1). Atkin conjectures that 389 is the largest prime such that the cusp of X_{0}^{+}(389) fails to be a Weierstrass point (see Section 26.4.2).

26.1 Factors of J_{0}(389)

To each newform f ∈ S_{2}(Γ_{0}(389)), Shimura [Shi73] associated a quotient A_{f} of J_{0}(389), and J_{0}(389) is isogeneous to the product ∏ A_{f}, where the product runs over the Gal(\overline{Q}/Q)-conjugacy classes of newforms. Moreover, because 389 is prime each factor A_{f} cannot be decomposed further up to isogeny, even over \overline{Q} (see [Rib75]).

26.1.1 Newforms of level 389

There are five Gal(\overline{Q}/Q)-conjugacy classes of newforms in S_{2}(Γ_{0}(389)). The first class corresponds to the unique elliptic curve of conductor 389, and its q-exansion begins

f_{1} = q − 2q^{2} − 2q^{3} + 2q^{4} − 3q^{5} + 4q^{6} − 5q^{7} + q^{9} + 6q^{10} + · · · .

The second has coefficients in the quadratic field Q(\sqrt{2}), and has q-expansion

f_{2} = q + \sqrt{2}q^{2} + (\sqrt{2} − 2)q^{3} − q^{5} + (−2\sqrt{2} + 2)q^{6} + · · · .

The third has coefficients in the cubic field generated by a root α of x^{3} − 4x − 2:

f_{3} = q + αq^{2} − αq^{3} + (α^{2} − 2)q^{4} + (−α^{2} + 1)q^{5} − α^{2}q^{6} + · · · .

The fourth has coefficients that generate the degree-six field defined by a root β of x^{6} + 3x^{5} − 2x^{4} − 8x^{3} + 2x^{2} + 4x − 1 and q-expansion

f_{4} = q + βq^{2} + (β^{5} + 3β^{4} − 2β^{3} − 8β^{2} + β + 2)q^{3} + · · · .

The fifth and final newform (up to conjugacy) has coefficients that generate the degree 20 field defined by a root of

f_{5} = x^{20} − 3x^{19} − 29x^{18} + 91x^{17} + 338x^{16} − 1130x^{15} − 2023x^{14} + 7432x^{13}
+6558x^{12} − 28021x^{11} − 10909x^{10} + 61267x^{9} + 6954x^{8} − 74752x^{7}
+1407x^{6} + 46330x^{5} − 1087x^{4} − 12558x^{3} − 942x^{2} + 960x + 148.

26.1 Factors of J_{0}(389) 259

FIGURE 26.1.1. Congruences Between Newforms

Congruences

The vertices in Figure 26.1.1 correspond to the newforms f_{i}; there is an edge between f_{i} and f_{j} labeled p if there is a maximal ideal ℘ | p of the field generated by the Fourier coefficients of f_{i} and f_{j} such that f_{i} ≡ f_{j} (mod ℘).

26.1.2 Isogeny structure

We deduce from the above determination of the newforms in S_{2}(Γ_{0}(389)) that J_{0}(389) is Q-isogenous to a product of \overline{Q}-simple abelian varieties

J ∼ A_{1} × A_{2} × A_{3} × A_{4} × A_{5}.

View the duals A_{i}^{∨} of the A_{i} as abelian subvarieties of J_{0}(389). Using modular symbols as in [AS05, §3.4] we find that, for i ≠ j, a prime p divides #(A_{i}^{∨} ∩ A_{j}^{∨}) if and only if f_{i} ≡ f_{j} (mod ℘) for some prime ℘ | p (recall that the congruence primes are given in Figure 26.1.1 above).

26.1.3 Mordell-Weil ranks

Suppose f ∈ S_{2}(Γ_{0}(N)) is a newform of some level N. The functional equation for L(f, s) implies that ord_{s=1} L(f, s) is odd if and only if the sign of the eigenvalue of the Atkin-Lehner involution W_{N} on f is +1.

Proposition 26.1.1. If f ∈ S_{2}(Γ_{0}(N)) is a newform of level N < 389, then ord_{s=1} L(f, s) is either0 or 1.

Proof. The proof amounts to a large computation, which divides into two parts:

1. Verify, for each newform f of level N < 389 such that W_{N}(f) = −f, that L(f, 1) = ∗ ∫_{0}^{i∞} f(z)dz (for some nonzero ∗) is nonzero. This is a purely algebraic computation involving modular symbols.

2. Verify, for each newform f of level N < 389 such that W_{N}(f) = f, that L′(f, 1) ≠ 0 (see [FpS^{+}01, §4.1], which points to [Cre97, §2.11,§2.13]). We do this by approximating an infinite series that converges to L′(f, 1) and noting that the value we get is far from 0.

Thus N = 389 is the smallest level such that the L-series of some factor A_{f} of J_{0}(N) has order of vanishing higher than that which is forced by the sign in the functional equation.

Proposition 26.1.2. The following table summarizes the dimensions and Mordell-Weil ranks (over the image of the Hecke ring) of the newform factors of J_{0}(N):

A_{1} A_{2} A_{3} A_{4} A_{5}
Dimension 1 2 3 6 20
Rank  2 1 1 1 0

Proof. The elliptic curve A_{1} is 389A in Cremona’s tables, which is the elliptic curve of smallest conductor having rank 2. For A_{5} we directly compute whether or not the L-function vanishes using modular symbols, by taking an inner product with the winding element e_{w} = −{0, ∞}. We find that the L-function does not vanish. By Kolyvagin-Logachev, it follows that A_{5} has Mordell-Weil rank 0.

For each of the other three factors, the sign of the functional equation is odd, so the analytic ranks are odd. As in the proof of Proposition 26.1.1, we verify that the analytic rank is 1 in each case. By work of Gross, Zagier, and Kolyvagin it follows that the ranks are 1.

26.2 The Hecke algebra

26.2.1 The Discriminant is divisible by p

Let N be a positive integer. The Hecke algebra T ⊂ End(S_{2}(Γ_{0}(N))) is the subring generated by all Hecke operators T_{n} for n = 1, 2, 3, . . .. We are concerned with the discriminant of the trace pairing (t, s) ↦ Tr(ts).

When N is prime, T_{Q} = T ⊗_{Z} Q is a product K_{1} × ⋯ × K_{n} of totally real number fields. Let T̃ denote the integral closure of T in T_{Q}; note that T̃ = ∏ O_{i} where O_{i} is the ring of integers of K_{i}. Then disc(T) = [T̃ : T] · ∏_{i=1}^{n} disc(K_{i}).

Proposition 26.2.1. The discriminant of the Hecke algebra associated to S_{2}(Γ_{0}(389)) is

2^{53} · 3^{4} · 5^{6} · 31^{2} · 37 · 389 · 3881 · 215517113148241 · 477439237737571441.

Proof. By [?], the Hecke algebra T is generated as a Z-module by T_{1}, T_{2}, … T_{65}. To compute disc(T), we proceed as follows. First, compute the space S_{2}(Γ_{0}(389)) of cuspidal modular symbols, which is a faithful T-module. Choose a random element x ∈ S_{2}(Γ_{0}(389))_{+} of the +1-quotient of the cuspidal modular symbols, then compute the images v_{1} = T_{1}(x), v_{2} = T_{2}(x), …, v_{65} = T_{65}(x). If these don’t span a space of dimension 32 = rank_{Z} T choose a new random element x and repeat. Using the Hermite Normal Form, find a Z-basis b_{1}, …, b_{32} for the Z-span of v_{1}, …, v_{65}. The trace pairing on T induces a trace pairing on the v_{i}, and hence on the b_{i}. Then disc(T) is the discriminant of this pairing on the b_{i}. The reason we embed T in S_{2}(Γ_{0}(389))_{+} as Tx is because directly finding a Z-basis for T would involve computing the Hermite Norm Form of a list of 65 vectors in a 1024dimensional space, which is unnecessarily difficult (though possible).

We compute this discriminant by applying the definition of discriminant to a matrix representation of the first 65 Hecke operators T_{1}, …, T_{65}. Matrices representing these Hecke operators were computed using the modular symbols algorithms described in [Cre97]. [Sturm, On the congruence of modular forms].

In the case of X_{0}(389), T ⊗ Q = K_{1} × K_{2} × K_{3} × K_{6} × K_{20}, where K_{d} has degree d over Q. We have

K_{1} = Q,
K_{2} = Q(\sqrt{2})
K_{3} = Q(β), β^{3} − 4β − 2 = 0,
K_{6} = Q(γ), γ^{6} + 3γ^{5} − 2γ^{4} − 8γ^{3} + 2γ^{2} + 4γ − 1 = 0,
K_{20} = Q(δ), δ^{20} − 3δ^{19} − 29δ^{18} + 91δ^{17} + 338δ^{16} − 1130δ^{15} − 2023δ^{14} + 7432δ^{13}

+6558δ^{12} − 28021δ^{11} − 10909δ^{10} + 61267δ^{9} + 6954δ^{8} − 74752δ^{7} +1407δ^{6} + 46330δ^{5} − 1087δ^{4} − 12558δ^{3} − 942δ^{2} + 960δ + 148 = 0.

The discriminants of the K_{i} are

\begin{matrix}K_{1} & K_{2} & K_{3} & K_{6} \\ 1 & 2^{3} & 2^{2} · 37 & 5^{3} · 3881 \end{matrix}

and

disc(K_{20}) = 2^{14} · 5 · 389 · 215517113148241 · 477439237737571441.

Observe that the discriminant of K_{20} is divisible by 389. The product of the discriminants is

2^{19} · 5^{4} · 37 · 389 · 3881 · 215517113148241 · 477439237737571441.

This differs from the exact discriminant by a factor of 2^{34} · 3^{4} · 5^{2} · 31^{2}, so the index of T in its normalization is

[T̃ : T] = 2^{17} · 3^{2} · 5 · 31.

Notice that 389 does not divide this index, and that 389 is not a “congruence prime”, so 389 does not divide any modular degrees.

Question 26.2.2. Is there a newform optimal quotient A_{f} of J_{0}(p) such that p divides the modular degree of A_{f}? (No, if p < 14000.)

26.2.2 Congruences primes in S_{p+1}(Γ_{0}(1))

K. Ono asked the following question, in connection with Theorem 1 of [?].

Question 26.2.3. Let p be a prime. Is p ever a congruence prime on S_{p+1}(Γ_{0}(1))? More precisely, if K is the number field generated by all the eigenforms of weight p + 1 on Γ_{0}(1), can there be a prime ideal ℘ | p for which f ≡ g (mod ℘) for distinct eigenforms f, g ∈ S_{p+1}(Γ_{0}(1))?

The answer is “yes”. There is a standard relationship between S_{p+1}(Γ_{0}(1)) and S_{2}(Γ_{0}(p)). As noted in Section 26.2.1, p = 389 is a congruence prime for S_{2}(Γ_{0}(389)), so we investigate S_{389+1}(Γ_{0}(1)).

Proposition 26.2.4. There exist distinct newforms f, g ∈ S_{389+1}(Γ_{0}(1)) and a prime ℘ of residue characteristic 389 such that f ≡ g (mod ℘).

Proof. We compute the characteristic polynomial f of the Hecke operator T_{2} on S_{389+1}(Γ_{0}(1)) using nothing more than [Ser73, Ch. VII]. We find that f factors modulo 389 as follows:

\overline{f} = (x + 2)(x + 56)(x + 135)(x + 158)(x + 175)^{2}(x + 315)(x + 342)(x^{2} + 387)
(x^{2} + 97x + 164)(x^{2} + 231x + 64)(x^{2} + 286x + 63)
(x^{5} + 88x^{4} + 196x^{3} + 113x^{2} + 168x + 349)
(x^{11} + 276x^{10} + 182x^{9} + 13x^{8} + 298x^{7} + 316x^{6} + 213x^{5}
+248x^{4} + 108x^{3} + 283x^{2} + x + 101)

Moreover, f is irreducible and 389 || disc(f), so the square factor (x+175)^{2} implies that 389 is ramified in the degree-32 field L generated by a single root of f. Thus there are exactly 31 distinct homomorphisms from the ring of integers of L to \overline{F}_{389}. That is, there are exactly 31 ways to reduce the q-expansion of a newform in S_{390}(Γ_{0}(1)) to obtain a q-expansion in \overline{F}_{389}[[q]]. Let K be the field generated by all eigenvalues of the 32 newforms g_{1}, … g_{32} ∈ S_{390}(Γ_{0}(1)), and let ℘ be a prime of O_{K} lying over 389. Then the subset {g_{1} (mod ℘), g_{2} (mod ℘), …, g_{32} (mod ℘)} of \overline{F}_{389}[[q]] has cardinality at most 31, so there exists i ≠ j such that g_{i} ≡ g_{j} (mod ℘).

26.3 Supersingular points in characteristic 389

26.3.1 The Supersingular j-invariants in characteristic 389

Let α be a root of α^{2} + 95α + 20. Then the 33 = g(X_{0}(389)) + 1 supersingular j-invariants in F_{389^{2}} are

0, 7, 16, 17, 36, 121, 154, 220, 318, 327, 358, 60α + 22, 68α + 166, 80α + 91, 86α + 273, 93α + 333, 123α + 350, 123α + 375, 129α + 247, 131α + 151, 160α + 321, 176α + 188, 213α + 195, 229α + 292, 258α + 154, 260α + 51, 266α + 335, 266α + 360, 296α + 56, 303α + 272, 309α + 271, 321α + 319, 329α + 157.

26.4 Miscellaneous

26.4.1 The Shafarevich-Tate group

Using visibility theory [AS02, §4.2], one sees that #� (A_{5}) is divisible by an odd prime, because

(Z/5Z)^{2} ≈ A_{1}(Q)/5A_{1}(Q) ⊂ � (A_{5}).

Additional computations suggest the following conjecture.

Conjecture 26.4.1. N = 389 is the smallest level such that there is an optimal newform quotient A_{f} of J_{0}(N) with #� (A_{f}) divisible by an odd prime.

26.4.2 Weierstrass points on X_{0}^{+}(p)

Oliver Atkin has conjectured that 389 is the largest prime such that the cusp on X_{0}^{+}(389) fails to be a Weierstrass point. He verified that the cusp of X_{0}^{+}(389) is not a Weierstrass point but that the cusp of X_{0}^{+}(p) is a Weierstrass point for all primes p such that 389 < p ≤ 883 (see, e.g., [Elk98, pg.39]). In addition, the author has extended the verification of Atkin’s conjecture for all primes < 3000. Explicitly, this involves computing a reduced-echelon basis for the subspace of S_{2}(Γ_{0}(p)) where the Atkin-Lehner involution W_{p} acts as +1, and comparing the largest valuation of an element of this basis with the dimension of the subspace. These numbers differ exactly when the cusp is a Weierstrass point.

26.4.3 A Property of the plus part of the integral homology

For any positive integer N, let H^{+}(N) = H_{1}(X_{0}(N), Z)^{+} be the +1 eigen-submodule for the action of complex conjugation on the integral homology of X_{0}(N). Then H^{+}(N) is a module over the Hecke algebra T. Let

F^{+}(N) = coker (H^{+}(N) × Hom(H^{+}(N), Z) → Hom(T, Z))

where the map sends (x, φ) to the homomorphism t ↦ φ(tx). Then #F^{+}(p) ∈ {1, 2, 4} for all primes p < 389, but #F^{+}(389) = 8.

26.4.4 The Field generated by points of small prime order on an elliptic curve

The prime 389 arises in a key way in the verification of condition 3 in [MS01].

264 26. The Modular Curve X_{0}(389)

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