Component Groups of Quotients of J_{0}(N)

David Kohel^{1} and William A. Stein^{2}

1 University of Sydney

                         kohel@maths.usyd.edu.au
                http://www.maths.usyd.edu.au:8000/u/kohel/

2 University of California at Berkeley,

                          was@math.berkeley.edu
                  http://shimura.math.berkeley.edu/~was

Abstract. Let f be a newform of weight 2 on Γ_{0}(N), and let A_{f} be the corresponding optimal abelian variety quotient of J_{0}(N). We describe an algorithm to compute the order of the component group of A_{f} at primes p that exactly divide N. We give a table of orders of component groups for all f of level N ≤ 127 and five examples in which the component group is very large, as predicted by the Birch and Swinnerton-Dyer conjecture.

1 Introduction

Let X_{0}(N) be the Riemann surface obtained by compactifying the quotient of the upper half-plane by the action of Γ_{0}(N). Then X_{0}(N) has a canonical structure of algebraic curve over Q; denote its Jacobian by J_{0}(N). It is equipped with an action of a commutative ring T = Z[… T_{n} …] of Hecke operators. For more details on modular curves, Hecke operators, and modular forms see, e.g., [8].

Now suppose that f = ∑_{n=1}^{∞} a_{n}q^{n} is a modular newform of weight 2 for the congruence subgroup Γ_{0}(N). The Hecke operators also act on f by T_{n}(f) = a_{n}f. The eigenvalues a_{n} generate an order R_{f} = Z[… a_{n} …] in a number field K_{f}. The kernel I_{f} of the map T → R_{f} sending T_{n} to a_{n} is a prime ideal. Following Shimura [15], we associate to f the quotient A_{f} = J_{0}(N)/I_{f}J_{0}(N) of J_{0}(N). Then A_{f} is an abelian variety over Q of dimension [K_{f} : Q], with bad reduction exactly at the primes dividing N.

One-dimensional quotients of J_{0}(N) have been intensely studied in recent years, both computationally and theoretically. The original conjectures of Birch and Swinnerton-Dyer [1, 2], for elliptic curves over Q, were greatly influenced by computations. The scale of these computations was extended and systematized by Cremona in [6].

In another direction, Wiles [20] and Taylor-Wiles [18] proved a special case of the conjecture of Shimura-Taniyama, which asserts that every elliptic curve over Q is a quotient of some J_{0}(N); this allowed them to establish Fermat’s Last Theorem. The full Shimura-Taniyama conjecture was later proved by Breuil, Conrad, Diamond, and Taylor in [4]. This illustrates the central role played by quotients of J_{0}(N).

2 Component Groups of A_{f}

The Néron model A/Z of an abelian variety A/Q is by definition a smooth commutative group scheme over Z with generic fiber A such that for any smooth scheme S over Z, the restriction map

Hom_{Z}(S, A) → Hom_{Q}(S_{Q}, A)

is a bijection. For more details, including a proof of existence, see, e.g., [5].

Suppose that A_{f} is a quotient of J_{0}(N) corresponding to a newform f on Γ_{0}(N), and let A_{f} be the Néron model of A_{f}. For any prime divisor p of N, the closed fiber A_{f}_{/F_{p}} is a group scheme over F_{p}, which need not be connected. Denote the connected component of the identity by A_{f}^{∘}_{/F_{p}}. There is an exact sequence

0 → A_{f}^{∘}_{/F_{p}} → A_{f}_{/F_{p}} → Φ_{A_{f} ,p} → 0

with Φ_{A_{f} ,p} a finite étale group scheme over F_{p} called the component group of A_{f} at p.

The category of finite étale group schemes over F_{p} is equivalent to the category of finite groups equipped with an action of Gal(\overline{F}_{p}/F_{p}) (see, e.g., [19, §6.4]). The order of an étale group scheme G/F_{p} is defined to be the order of the group G(\overline{F}_{p}). In this paper we describe an algorithm for computing the order of Φ_{A_{f} ,p}, when p exactly divides N.

3 The Algorithm

Let J = J_{0}(N), fix a newform f of weight-two for Γ_{0}(N), and let A_{f} be the corresponding quotient of J. Because J is the Jacobian of a curve, it is canonically isomorphic to its dual, so the projection J → A_{f} induces a polarization A_{f}^{∨} → A_{f}, where A_{f}^{∨} denotes the abelian variety dual of A_{f}. We define the modular degree δ_{A_{f}} of A_{f} to be the positive square root of the degree of this polarization. This agrees with the usual notion of modular degree when A_{f} is an elliptic curve.

A torus T over a field k is a group scheme whose base extension to the separable closure k_{s} of k is a finite product of copies of G_{m}. Every commutative algebraic group over k admits a unique maximal subtorus, defined over k, whose formation commutes with base extension (see IX §2.1 of [9]). The character group of a torus T is the group X = Hom_{k_{s}}(T, G_{m}) which is a free abelian group of finite rank together with an action of Gal(k_{s}/k) (see, e.g., [19, §7.3]).

We apply this construction to our setting as follows. The closed fiber of the Néron model of J at p is a group scheme over F_{p}, whose maximal torus we denote by T_{J,p}. We define X_{J,p} to be the character group of T_{J,p}. Then X_{J,p} is a free abelian group equipped with an action of both Gal(\overline{F}_{p}/F_{p}) and the Hecke algebra T (see, e.g., [14]). Moreover, there exists a bilinear pairing

⟨ , ⟩ : X_{J,p} × X_{J,p} → Z

called the monodromy pairing such that

Φ_{J,p} ≅ coker(X_{J,p} → Hom(X_{J,p}, Z)).

Let X_{J,p}[I_{f}] be the intersection of all kernels ker(t) for t in I_{f}, and let

α_{f} : X_{J,p} → Hom(X_{J,p}[I_{f}], Z)

be the map induced by the monodromy pairing. The following theorem of the second author [16], provides the basis for the computation of orders of component groups.

Theorem 1. With the notation as above, we have the equality

#Φ_{A_{f} ,p} = \frac{#coker(α_{f}) · δ_{A_{f}}}{#(α_{f}(X_{J,p})/α_{f}(X_{J,p}[I_{f}]))} .

3.1 Computing the modular degree δ_{A,f}

Using modular symbols (see, e.g., [6]), we first compute the homology group H_{1}(X_{0}(N), Q; cusps). Using lattice reduction, we compute the Zsubmodule H_{1}(X_{0}(N), Z; cusps) generated by all Manin symbols (c, d). Then H_{1}(X_{0}(N), Z) is the integer kernel of the boundary map.

The Hecke ring T acts on H_{1}(X_{0}(N), Z) and also on the linear dual Hom(H_{1}(X_{0}(N), Z), Z), where t ∈ T acts on φ ∈ Hom(H_{1}(X_{0}(N), Z), Z) by (t.φ)(x) = φ(tx). We have a natural restriction map

r_{f} : Hom(H_{1}(X_{0}(N), Z), Z)[I_{f}] → Hom(H_{1}(X_{0}(N), Z)[I_{f}], Z).

Proposition 1. The cokernel of r_{f} is isomorphic to the kernel of the polarization A_{f}^{∨} → A_{f} induced by the map J_{0}(N) → A_{f}.

Thus the order of the cokernel of r_{f} is the square of the modular degree δ_{f}. We pause to note that the degree of any polarization is a square; see, e.g., [13, Thm. 13.3].

Proof. Let S = S_{2}(Γ_{0}(N), C) be the complex vector space of weight-two modular forms of level N, and set H = H_{1}(X_{0}(N), Z). The integration pairing S × H → C induces a natural map

Φ_{f} : H → Hom(S[I_{f}], C).

Using the classical Abel-Jacobi theorem, we deduce the following commutative diagram, which has exact columns, but whose rows are not exact.

\xymatrix{0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ H[I_{f}] \ar[d] \ar[r] & H \ar[d] \ar[r] & Φ_{f}(H) \ar[d] \\ Hom(S, C)[I_{f}] \ar[d] \ar[r] & Hom(S, C) \ar[d] \ar[r] & Hom(S[I_{f}], C) \ar[d] \\ A_{f}^{∨}(C) \ar[d] \ar[r] \ar[rr] & J_{0}(N)(C) \ar[d] \ar[r] & A_{f}(C) \ar[d] \\ 0 & 0 & 0}

By the snake lemma, the kernel of A_{f}^{∨}(C) → A_{f}(C) is isomorphic to the cokernel of the map H[I_{f}] → Φ_{f}(H). Since

Hom(H/ ker(Φ_{f}), Z) ≅ Hom(H, Z)[I_{f}],

the Hom(−, Z) dual of the map H[I_{f}] → Φ_{f}(H) = H/ ker(Φ_{f}) is r_{f}, which proves the proposition.

3.2 Computing the character group X_{J,p}

Let N = Mp, where M and p are coprime. If M is small, then the algorithm of Mestre and Oesterlé [12] can be used to compute X_{J,p}. This algorithm constructs the graph of isogenies between \overline{F}_{p}-isomorphism classes of pairs consisting of a supersingular elliptic curve and a cyclic M-torsion subgroup. In particular, the method is elementary to apply when X_{0}(M) has genus 0.

In general, the above category of “enhanced” supersingular elliptic curves can be replaced by one of left (or right) ideals of a quaternion order O of level M in the quaternion algebra over Q ramified at p. This gives an equivalent category, in which the computation of homomorphisms is efficient. The character group X_{J,p} is known by Deligne-Rapoport [7] to be canonically isomorphic to the degree zero subgroup X (O) of the free abelian “divisor group” on the isomorphism classes of enhanced supersingular elliptic curves and of quaternion ideals. Moreover, this isomorphism is compatible with the operation of Hecke operators, which are effectively computable in X (O) in terms of ideal homomorphisms.

The inner product of two classes in this setting is defined to be the number of isomorphisms between any two representatives. The linear extension to X (O) gives an inner product which agrees, under the isomorphism, with the monodromy pairing on X_{J,p}. This gives, in particular, an isomorphism Φ_{J,p} ≅ coker(X (O) → Hom(X (O), Z)), and an effective means of computing #coker(α_{f}) and #(α_{f}(X_{J,p})/α_{f}(X_{J,p}[I_{f}])).

The arithmetic of quaternions has been implemented in Magma [11] by the first author. Additional details and the application to Shimura curves, generalizing X_{0}(N), can be found in Kohel [10].

3.3 The Galois action on Φ_{A_{f},p}

To determine the Galois action on Φ_{A_{f} ,p}, we need only know the action of the Frobenius automorphism Frob_{p}. However, Frob_{p} acts on Φ_{A_{f} ,p} in the same way as −W_{p}, where W_{p} is the pth Atkin-Lehner involution, which can be computed using modular symbols. Since f is an eigenform, the involution W_{p} acts as either +1 or −1 on Φ_{A_{f} ,p}. Moreover, the operator W_{p} is determined by an involution on the set of quaternion ideals, so it can be determined explicitly on the character group.

4 Tables

The main computational results of this work are presented below in two tables. The relevant algorithms have been implemented in Magma and will be made part of a future release. They can also be obtained from the second author.

4.1 Component groups at low level

Table 4.1 gives the component groups of the quotients A_{f} of J_{0}(N) for N ≤ 127. The column labeled d contains the dimensions of the A_{f}, and the column labeled #Φ_{A_{f} ,p} contains a list of the orders of the component groups of A_{f}, one for each divisor p of N, ordered by increasing p. An entry of “?” indicates that p^{2} | N, so our algorithm does not apply. A component group order is starred if the Gal(\overline{F}_{p}/F_{p})-action is nontrivial. More data along these lines can be obtained from the second author.

4.2 Examples of large component groups

Let Ω_{A_{f}} be the real period of A_{f}, as defined by J. Tate in [17]. The second author computed the rational numbers L(A_{f}, 1)/Ω_{A_{f}} for every newform f of level N ≤ 1500. The five largest prime divisors occur in the ratios given in Table 4.2. The Birch and Swinnerton-Dyer conjecture predicts that the large prime divisor of the numerator of each special value must divide the order either of some component group Φ_{A_{f} ,p} or of the Shafarevich-Tate group of A_{f}. In each instance Φ_{A_{f} ,2} is divisible by the large prime divisor, as predicted.

5 Further directions

Further considerations are needed to compute the group structure of Φ_{A_{f} ,p}. However, since the action of Frobenius is known, computing the group structure of Φ_{A_{f} ,p} suffices to determine its structure as a group scheme.

Our methods say nothing about the component group at primes whose square divides the level. The free abelian group on classes of nonmaximal orders of index p at a ramified prime gives a well-defined divisor group. Do the resulting Hecke modules determine the component groups for quotients of level p^{2}M?

Is it possible to define quantities as in Theorem 1 even when the weight of f is greater than 2? If so, how are the resulting quantities related to the Bloch-Kato Tamagawa numbers (see [3]) of the higher weight motive attached to f?

Component groups at low level

\begin{aligned}N & d & #Φ_{A_{f} ,p} \\ 11 & 1 & 5 \\ 14 & 1 & 6^{∗}, 3 \\ 15 & 1 & 4^{∗}, 4 \\ 17 & 1 & 4 \\ 19 & 1 & 3 \\ 20 & 1 & ?, 2^{∗} \\ 21 & 1 & 4, 2^{∗} \\ 23 & 2 & 11 \\ 24 & 1 & ?, 2^{∗} \\ 26 & 1 & 3^{∗}, 3 \\ & 1 & 7, 1^{∗} \\ 27 & 1 & ? \\ 29 & 2 & 7 \\ 30 & 1 & 4^{∗}, 3, 1^{∗} \\ 31 & 2 & 5 \\ 32 & 1 & ? \\ 33 & 1 & 6^{∗}, 2 \\ 34 & 1 & 6, 1^{∗} \\ 35 & 1 & 3^{∗}, 3 \\ & 2 & 8, 4^{∗} \\ 36 & 1 & ?,? \\ 37 & 1 & 1^{∗} \\ & 1 & 3 \\ 38 & 1 & 9^{∗}, 3 \\ & 1 & 5, 1^{∗} \\ 39 & 1 & 2^{∗}, 2 \\ & 2 & 14, 2^{∗} \\ 40 & 1 & ?, 2 \\ 41 & 3 & 10 \\ 42 & 1 & 8, 2^{∗}, 1^{∗} \\ 43 & 1 & 1^{∗} \\ & 2 & 7 \\ 44 & 1 & ?, 1^{∗} \\ 45 & 1 & ?, 1^{∗} \\ 46 & 1 & 10^{∗}, 1 \\ 47 & 4 & 23 \\ 48 & 1 & ?, 2 \\ 49 & 1 & ? \\ 50 & 1 & 1^{∗},? \\ & 1 & 5,? \\ 51 & 1 & 3, 1^{∗} \\ & 2 & 16^{∗}, 4 \\ 52 & 1 & ?, 2^{∗} \\ 53 & 1 & 1^{∗} \end{aligned}  \begin{aligned}N & d & #Φ_{A_{f} ,p} \\ & 3 & 13 \\ 54 & 1 & 3^{∗},? \\ & 1 & 3,? \\ 55 & 1 & 2, 2^{∗} \\ & 2 & 14^{∗}, 2 \\ 56 & 1 & ?, 1 \\ & 1 & ?, 1^{∗} \\ 57 & 1 & 2^{∗}, 1^{∗} \\ & 1 & 2, 2^{∗} \\ & 1 & 10, 1^{∗} \\ 58 & 1 & 2^{∗}, 1^{∗} \\ & 1 & 10, 1^{∗} \\ 59 & 5 & 29 \\ 61 & 1 & 1^{∗} \\ & 3 & 5 \\ 62 & 1 & 4, 1^{∗} \\ & 2 & 66^{∗}, 3 \\ 63 & 1 & ?, 1^{∗} \\ & 2 & ?, 3 \\ 64 & 1 & ? \\ 65 & 1 & 1^{∗}, 1^{∗} \\ & 2 & 3^{∗}, 3 \\ & 2 & 7, 1^{∗} \\ 66 & 1 & 2^{∗}, 3, 1^{∗} \\ & 1 & 4, 1^{∗}, 1^{∗} \\ & 1 & 10, 5, 1 \\ 67 & 1 & 1 \\ & 2 & 1^{∗} \\ & 2 & 11 \\ 68 & 2 & ?, 2^{∗} \\ 69 & 1 & 2, 1^{∗} \\ & 2 & 22^{∗}, 2 \\ 70 & 1 & 4, 2^{∗}, 1^{∗} \\ 71 & 3 & 5 \\ & 3 & 7 \\ 72 & 1 & ?,? \\ 73 & 1 & 2 \\ & 2 & 1^{∗} \\ & 2 & 3 \\ 74 & 2 & 9^{∗}, 3 \\ & 2 & 95, 1^{∗} \\ 75 & 1 & 1^{∗},? \\ & 1 & 1,? \\ & 1 & 5,? \end{aligned} \begin{aligned}N & d & #Φ_{A_{f} ,p} \\ 76 & 1 & ?, 1^{∗} \\ 77 & 1 & 2^{∗}, 1^{∗} \\ & 1 & 3^{∗}, 2 \\ & 1 & 6, 3^{∗} \\ & 2 & 2, 2^{∗} \\ 78 & 1 & 16^{∗}, 5^{∗}, 1 \\ 79 & 1 & 1^{∗} \\ & 5 & 13 \\ 80 & 1 & ?, 2 \\ & 1 & ?, 2^{∗} \\ 81 & 2 & ? \\ 82 & 1 & 2^{∗}, 1^{∗} \\ & 2 & 28, 1^{∗} \\ 83 & 1 & 1^{∗} \\ & 6 & 41 \\ 84 & 1 & ?, 1^{∗}, 2^{∗} \\ & 1 & ?, 3, 2 \\ 85 & 1 & 2^{∗}, 1 \\ & 2 & 2^{∗}, 1^{∗} \\ & 2 & 6, 1^{∗} \\ 86 & 2 & 21^{∗}, 3 \\ & 2 & 55, 1^{∗} \\ 87 & 2 & 5, 1^{∗} \\ & 3 & 92^{∗}, 4 \\ 88 & 1 & ?, 1^{∗} \\ & 2 & ?, 2^{∗} \\ 89 & 1 & 1^{∗} \\ & 1 & 2 \\ & 5 & 11 \\ 90 & 1 & 2^{∗},?, 3 \\ & 1 & 6,?, 1^{∗} \\ & 1 & 4,?, 1 \\ 91 & 1 & 1^{∗}, 1^{∗} \\ & 1 & 1, 1 \\ & 2 & 7, 1^{∗} \\ & 3 & 4^{∗}, 8 \\ 92 & 1 & ?, 1^{∗} \\ & 1 & ?, 1 \\ 93 & 2 & 4^{∗}, 1^{∗} \\ & 3 & 64, 2^{∗} \\ 94 & 1 & 2, 1^{∗} \\ & 2 & 94^{∗}, 1 \\ 95 & 3 & 10, 2^{∗} \\ & 4 & 54^{∗}, 6 \end{aligned} \begin{aligned}N & d & #Φ_{A_{f} ,p} \\ 96 & 1 & ?, 2 \\ & 1 & ?, 2^{∗} \\ 97 & 3 & 1^{∗} \\ & 4 & 8 \\ 98 & 1 & 2^{∗},? \\ & 2 & 14,? \\ 99 & 1 & ?, 1^{∗} \\ & 1 & ?, 1 \\ & 1 & ?, 1^{∗} \\ & 1 & ?, 1^{∗} \\ 100 & 1 & ?,? \\ 101 & 1 & 1^{∗} \\ & 7 & 25 \\ 102 & 1 & 2^{∗}, 2^{∗}, 1^{∗} \\ & 1 & 6^{∗}, 6, 1^{∗} \\ & 1 & 8, 4, 1 \\ 103 & 2 & 1^{∗} \\ & 6 & 17 \\ 104 & 1 & ?, 1^{∗} \\ & 2 & ?, 2 \\ 105 & 1 & 1, 1, 1 \\ & 2 & 10^{∗}, 2^{∗}, 2 \\ 106 & 1 & 4^{∗}, 1^{∗} \\ & 1 & 5^{∗}, 1 \\ & 1 & 24, 1^{∗} \\ & 1 & 3, 1^{∗} \\ 107 & 2 & 1^{∗} \\ & 7 & 53 \\ 108 & 1 & ?,? \\ 109 & 1 & 1 \\ & 3 & 1^{∗} \\ & 4 & 9 \\ 110 & 1 & 7^{∗}, 1^{∗}, 3 \\ & 1 & 3, 1^{∗}, 1^{∗} \\ & 1 & 5, 5, 1 \\ & 2 & 16^{∗}, 3, 1^{∗} \\ 111 & 3 & 10^{∗}, 2 \\ & 4 & 266, 2^{∗} \\ 112 & 1 & ?, 1^{∗} \\ & 1 & ?, 1 \\ & 1 & ?, 1^{∗} \\ 113 & 1 & 2 \\ & 2 & 2 \\ & 3 & 1^{∗} \end{aligned}  \begin{aligned}N & d & #Φ_{A_{f} ,p} \\ & 3 & 7 \\ 114 & 1 & 2^{∗}, 5^{∗}, 1 \\ & 1 & 20, 3^{∗}, 1^{∗} \\ & 1 & 6, 3, 1 \\ 115 & 1 & 5^{∗}, 1 \\ & 2 & 4^{∗}, 1^{∗} \\ & 4 & 32, 4^{∗} \\ 116 & 1 & ?, 1^{∗} \\ & 1 & ?, 2^{∗} \\ & 1 & ?, 1^{∗} \\ 117 & 1 & ?, 1 \\ & 2 & ?, 3 \\ & 2 & ?, 1^{∗} \\ 118 & 1 & 2^{∗}, 1^{∗} \\ & 1 & 19^{∗}, 1 \\ & 1 & 10, 1^{∗} \\ & 1 & 1, 1^{∗} \\ 119 & 4 & 9, 3^{∗} \\ & 5 & 48^{∗}, 8 \\ 120 & 1 & ?, 1, 1^{∗} \\ & 1 & ?, 2, 1 \\ 121 & 1 & ? \\ & 1 & ? \\ & 1 & ? \\ & 1 & ? \\ 122 & 1 & 4^{∗}, 1^{∗} \\ & 2 & 39^{∗}, 3 \\ & 3 & 248, 1^{∗} \\ 123 & 1 & 1^{∗}, 1^{∗} \\ & 1 & 5, 1 \\ & 2 & 7, 1^{∗} \\ & 3 & 184^{∗}, 4 \\ 124 & 1 & ?, 1^{∗} \\ & 1 & ?, 1 \\ 125 & 2 & ? \\ & 2 & ? \\ & 4 & ? \\ 126 & 1 & 8^{∗},?, 1^{∗} \\ & 1 & 2,?, 1 \\ 127 & 3 & 1^{∗} \\ & 7 & 21 \end{aligned}

Large L(A_{f} , 1)/Ω_{A_{f}}

\begin{matrix}N & dim & L(A_{f} , 1)/Ω_{A_{f}} & #Φ_{A_{f} ,p} \\ 1154 = 2·577 & 20 & 2^{?}·85495047371/17^{2} & 2^{?}·17^{2}·85495047371, 2^{?} \\ 1238 = 2·619 & 19 & 2^{?}·7553329019/5·31 & 2^{?}·5·31·7553329019, 2^{?} \\ 1322 = 2·661 & 21 & 2^{?}·57851840099/331 & 2^{?}·331·57851840099, 2^{?} \\ 1382 = 2·691 & 20 & 2^{?}·37·1864449649/173 & 2^{?}·37·173·1864449649, 2^{?} \\ 1478 = 2·739 & 20 & 2^{?}·7·29·1183045463/5·37 & 2^{?}·5·7·29·37·1183045463, 2^{?} \end{matrix}

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