Conjectures About Discriminants of Hecke Algebras of Prime Level

Frank Calegari^{⋆ ⋆ ⋆}^{1} and William A. Stein^{†}^{2}

1 Harvard University

                    fcale@math.harvard.edu
              http://www.math.harvard.edu/~fcale

2 Harvard University,

                     was@math.harvard.edu
               http://modular.fas.harvard.edu/

Abstract. In this paper, we study p-divisibility of discriminants of Hecke algebras associated to spaces of cusp forms of prime level. By considering cusp forms of weight bigger than 2, we are are led to make a precise conjecture about indexes of Hecke algebras in their normalisation which implies (if true) the surprising conjecture that there are no mod p congruences between non-conjugate newforms in S_{2}(Γ_{0}(p)), but there are almost always many such congruences when the weight is bigger than 2.

1 Basic Definitions

We first recall some commutative algebra related to discriminants, then introduce Hecke algebras of spaces of cusp forms.

1.1 Commutative Algebra

In this section we recall the definition of discriminant of a finite algebra and note that the discriminant is nonzero if and only if no base extension of the algebra contains nilpotents.

Let R be a ring and let A be an R-algebra that is free of finite rank as an R-module. The trace of x ∈ A is the trace, in the sense of linear algebra, of left multiplication by x.

Definition 1 (Discriminant). Let ω_{1}, …, ω_{n} be an R-basis for A. Then the discriminant disc(A) of A is the determinant of the n × n matrix (tr(ω_{i}ω_{j})).

⋆ ⋆ ⋆ Supported in part by the American Institute of Mathematics † Supported in part by a National Science Foundation Postdoctoral Fellowship

The discriminant is only well-defined modulo squares of units in R. When R = Z the discriminant is well defined, since the only units are ±1.

We say that A is separable over R if for every extension R′ of R, the ring A ⊗ R′ contains no nilpotents.

Proposition 1. Suppose R is a field. Then A has nonzero discriminant if and only if A is separable over R.

Proof. For the convenience of the reader, we summarize the proof in [Mat86, §26]. If A contains a nilpotent then that nilpotent is in the kernel of the trace pairing, so the discriminant is 0. Conversely, if A is separable then we may assume that R is algebraically closed. Then A is an Artinian reduced ring, hence isomorphic as a ring to a finite product of copies of R, since R is algebraically closed. Thus the trace form on A is nondegenerate.

1.2 The Discriminant Valuation

We next introduce Hecke algebras attached to certain spaces of cusp forms of prime level p, define the discriminant valuation as the exponent of the largest power of p that divides the discriminant, and observe that there are eigenform congruences modulo p exactly when the discriminant valuation is positive. We then present an example to illustrate the definitions.

Let Γ be a congruence subgroup of SL_{2}(Z). In this paper, we will only consider Γ = Γ_{0}(p) for p prime. For any positive integer k, let S_{k}(Γ) denote the space of holomorphic weight k cusp forms for Γ. Let

T = Z[…, T_{n}, …] ⊂ End(S_{k}(Γ))

be the associated Hecke algebra, which is generated by Hecke operators T_{n} for all integers n, including n = p (we will sometimes write U_{p} for T_{p}). Then T is a commutative ring that is free as a module over Z of rank equal to dim S_{k}(Γ). We will also sometimes consider the image T^{new} of T in End(S_{k}(Γ)^{new}).

Definition 2 (Discriminant Valuation). Let p be a prime, k a positive integer, and suppose that Γ = Γ_{0}(p). Let T be the corresponding Hecke algebra. Then the discriminant valuation of Γ in weight k is

d_{k}(Γ) = ord_{p}(disc(T)).

We expect that d_{k}(Γ) is finite for the following reason. The Hecke operators T_{n}, with n not divisible by p, are diagonalizable since they are self adjoint with respect to the Petersson inner product. When k = 2 one knows that U_{p} is diagonalizable since the level is square free, and when k > 2 one expects this (see [CE98]). If T contains no nilpotents, Proposition 1 implies that the discriminant of T is nonzero. Thus d_{k}(Γ) is finite when k = 2 and conjectured to be finite when k > 2.

Let p be a prime and suppose that Γ = Γ_{0}(p). A normalised eigenform is an element f = ∑ a_{n}q^{n} ∈ S_{k}(Γ) that is an eigenvector for all Hecke operators T_{ℓ}, including those that divide p, normalised so that a_{1} = 1. The quantity d_{k}(Γ) is of interest because it measures mod p congruences between normalised eigenforms in S_{k}(Γ).

Proposition 2. Assume that d_{k}(Γ) is finite. The discriminant valuation d_{k}(Γ) is positive (i.e., the discriminant is divisible by p) if and only if there is a congruence in characteristic p between two normalized eigenforms in S_{k}(Γ). (The two congruent eigenforms might be Galois conjugate.)

Proof. It follows from Proposition 1 that d_{k}(Γ) > 0 if and only if T ⊗ \overline{F}_{p} is not separable. The Artinian ring T ⊗ \overline{F}_{p} is not separable if and only if the number of ring homomorphisms T ⊗ \overline{F}_{p} → \overline{F}_{p} is less than

dim_{\overline{F}_{p}} T ⊗ \overline{F}_{p} = dim_{C} S_{k}(Γ).

Since d_{k}(Γ) is finite, the number of ring homomorphisms T ⊗ \overline{Q}_{p} → \overline{Q}_{p} equals dim_{C} S_{k}(Γ). The proposition follows from the fact that for any ring R, there is a bijection between ring homomorphisms T → R and normalised eigenforms with q-expansion in R.

The same proof also shows that a prime ℓ divides the discriminant of T if and only if there is a congruence mod ℓ between two normalized eigenforms in S_{k}(Γ)

Example 1. If Γ = Γ_{0}(389) and k = 2, then dim_{C} S_{2}(Γ) = 32. Let f be the characteristic polynomial of T_{2}. One can check that f is square free and 389 exactly divides the discriminant of f. This implies that d_{2}(Γ) = 1 and that T_{2} generates T ⊗ Z_{389} as an algebra over Z_{389}. (If T_{2} only generated a subring of T ⊗ Z_{389} of finite index > 1, then the discriminant of f would be divisible by 389^{2}.)

Modulo 389 the characteristic polynomial f is congruent to

(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)

The factor (x + 175)^{2} indicates that T ⊗ F_{389} is not separable over F_{389} since the image of (\overline{f}/(x+175))(T_{2}) in T⊗F_{389} is nilpotent (it is nonzero but its square is 0). There are 32 eigenforms over Q_{2} but only 31 mod 389 eigenforms, so there must be a congruence. There is a newform F in S_{2}(Γ_{0}(389), \overline{Z}_{389}) whose a_{2} term is a root of

x^{2} +(−39+190·389+96·389^{2} +⋯)x+(−106+43·389+19·389^{2} +⋯).

There is a congruence between F and its Gal(\overline{Q}_{389}/Q_{389})-conjugate.

2 Computing Discriminants

In this section we sketch the algorithm that we use for computing the discriminants mentioned in this paper.

This algorithm was inspired by a discussion of the second author with Hendrik Lenstra. We leave the details of converting the description below into standard matrix operations to the reader. Also, the modular symbols algorithms needed to compute Hecke operators are quite involved.

Let Γ = Γ_{0}(p), and let k ≥ 2 be an integer. The following sketches an algorithm for computing the discriminant of the Hecke algebra T acting on S_{k}(Γ).

1. For any given n, we can explicitly compute a matrix that represents the action of Hecke operators T_{n} on S_{k}(Γ) using modular symbols. We use the second author’s MAGMA [BCP97] packages for computing with modular symbols, which builds on work of many people (including [Cre97] and [Mer94]).

2. Using the Sturm bound, as described in the appendix to [LS02], find an integer b such that T_{1}, …, T_{b} generate T as a Z-module. (The integer b is ⌈(k/12) · [SL_{2}(Z) : Γ]⌉.)

3. Find a subset B of the T_{i} that form a Q-basis for T ⊗_{Z} Q. (This uses Gauss elimination.)

4. View T as a ring of matrices acting on Q^{d}, where d = dim(S_{k}(Γ)) and try random sparse vectors v ∈ Q^{d} until we find one such that the set of vectors C = {T (v) : T ∈ B} are linearly independent.

5. Write each of T_{1}(v), …, T_{b}(v) as Q-linear combinations of the elements of C.

6. Find a Z-basis D for the Z-span of these Q-linear combinations of elements of C. (This basis D corresponds to a Z-basis for T, but is much easier to find that directly looking for a Z-basis in the space of d × d matrices that T is naturally computed in.)

7. Unwinding what we have done in the previous steps, find the trace pairing on the elements of D, and deduce the discriminant of T by computing the determinant of the trace pairing matrix.

A very time-consuming step, at least in our implementation, is computing D from T_{1}(v), …, T_{b}(v) expressed in terms of C, and this explains why we embed T in Q^{d} instead of viewing the elements of T as vectors in Q^{d×d}.

An implementation by the second author of the above algorithm is included with the MAGMA computer algebra system. The relevant source

code is in the file Geometry/ModSym/linalg.m in the package directory

(or ask the second author of the apper to send you a copy linalg.m). We illustrate the use of MAGMA to compute discriminants below, which were run under MAGMA V2.10-21 for Linux on a computer with an Athlon 2800MP processor (2.1Ghz).

  > M := ModularSymbols(389,2, +1);
  > S := CuspidalSubspace(M);
  > time D  := DiscriminantOfHeckeAlgebra(S);
  Time: 0.750
  > D;
  629670054720061882880174736321392595498204931550235108311\
  04000000
  > Factorisation(D);
  [ <2, 53>, <3, 4>, <5, 6>, <31, 2>, <37, 1>, <389, 1>, ...]
  > M := ModularSymbols(997,2, +1); S := CuspidalSubspace(M);
  > time D  := DiscriminantOfHeckeAlgebra(S);
  Time: 55.600

The reason for the +1 in the construction of modular symbols is so that we compute on a space that is isomorphic as a T-module to one copy of S_{2}(Γ_{0}(p)), instead of two copies.

3 Data About Discriminant Valuations

In this section we report on our extensive computations of d_{k}(Γ_{0}(p)). We first note that there is only one p < 50000 such that d_{2}(Γ_{0}(p)) > 0.

Next we give a table of values of d_{4}(Γ_{0}(p)), which seems to exhibit a nice pattern.

3.1 Weight Two

Theorem 1. The only prime p < 60000 such that d_{2}(Γ_{0}(p)) > 0 is p = 389, with the possible exception of 50923 and 51437.

Computations in this direction by the second author have been cited in [Rib99], [MS01], [OW02], and [MO02]. For example, Theorem 1 is used for p < 1000 in [MS01] as a crucial step in proving that if E is an elliptic curve over Q(μ_{p}), with 17 ≤ p < 1000, then not all elements of E(\overline{Q})[p] are rational over Q(μ_{p}).

Proof. This is the result of a large computer computation. The rest of this proof describes how we did the computation, so the reader has some idea how to replicate or extend the computation. The computation described below took about one week using a cluster equipped with 10 Athlon 2000MP processors. The computations are nontrivial; we compute spaces of modular symbols, supersingular points, and Hecke operators on spaces of dimensions up to 5000.

The aim is to determine whether or not p divides the discriminant of the Hecke algebra of level p for each p < 60000. If T is an operator with integral characteristic polynomial, we write disc(T) for disc(charpoly(T)), which also equals disc(Z[T]). We will often use that

disc(T) mod p = disc(charpoly(T) mod p).

We ruled out the possibility that d_{k}(Γ_{0}(p)) > 0 for most levels p < 60000 by computing characteristic polynomials of Hecke operators using an algorithm that the second author and D. Kohel implemented in MAGMA ([BCP97]), which is based on the Mestre-Oesterle method of graphs [Mes86] (or contact the second author for an English translation). Our implementation is available as the “Module of Supersingular Points” package that comes with MAGMA. We computed disc(T_{q}) modulo p for several small primes q, and in most cases found a prime q such that this discriminant is nonzero. The following table summarises how often we used each prime q (note that there are 6057 primes up to 60000):

q number of p < 60000 where q smallest s.t. disc(T_{q}) ≠ 0 mod p
2 5809 times
3 161 (largest: 59471)
5 43 (largest: 57793)
7 15 (largest: 58699)
11 15 (the smallest is 307; the largest 50971)
13 2 (they are 577 and 5417)
17 3 (they are 17209, 24533, and 47387)
19 1 (it is 15661 )

The numbers in the right column sum to 6049, so 8 levels are missing. These are

389, 487, 2341, 7057, 15641, 28279, 50923, and 51437.

(The last two are still being processed. 51437 has the property that disc(T_{q}) = 0 for q = 2, 3, …, 17.) We determined the situation with the remaining 6 levels using Hecke operators T_{n} with n composite.

p  How we rule level p out, if possible
389 p does divide discriminant
487 using charpoly(T_{12})
2341 using charpoly(T_{6})
7057 using charpoly(T_{18})
15641 using charpoly(T_{6})
28279 using charpoly(T_{34})

Computing T_{n} with n composite is very time consuming when p is large, so it is important to choose the right T_{n} quickly. For p = 28279, here is a trick we used to quickly find an n such that disc(T_{n}) is not divisible by p. This trick might be used to speed up the computation for some other levels. The key idea is to efficiently discover which T_{n} to compute. Computing T_{n} on the full space of modular symbols is difficult, but using projections we can compute T_{n} on subspaces of modular symbols with small dimension more quickly (see, e.g., [Ste00, §3.5.2]). Let M be the space of mod p modular symbols of level p = 28279, and let f = gcd(charpoly(T_{2}), deriv(charpoly(T_{2}))). Let V be the kernel of f(T_{2}) (this takes 7 minutes to compute). If V = 0, we would be done, since then disc(T_{2}) ≠ 0 ∈ F_{p}. In fact, V has dimension 7. We find the first few integers n so that the charpoly of T_{n} on V has distinct roots, and they are n = 34, 47, 53, and 89. We then computed charpoly(T_{34}) directly on the whole space and found that it has distinct roots modulo p.

3.2 Some Data About Weight 4

The following are the valuations d = d_{4}(Γ_{0}(p)) at p of the discriminant of the Hecke algebras associated to S_{4}(Γ_{0}(p)) for p < 500. This data suggests a pattern, which motivates Conjecture 1 below.

p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
d 0 0 0 0 0 2 2 2 2 4 4 6 6 6 6 8 8
p 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139
d 10 10 10 12 12 12 14 16 16 16 16 18 18 20 20 22 24
p 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233
d 24 24 26 26 26 28 28 30 30 32 32 32 34 36 36 38 38
p 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337
d 38 40 40 42 42 44 44 46 46 46 48 50 50 52 52 54 56
p 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439
d 56 58 58 58 60 62 62 62 65 66 66 68 68 70 70 72 72
p 443 449 457 461 463 467 479 487 491 499
d 72 74 76 76 76 76 78 80 80 82

4 Speculations

Motivated by the promise of a pattern suggested by the table in Section 3.2, we computed d_{k}(Γ_{0}(p)) for many values of k and p. Our observations led us to the following results and conjectures.

Theorem 2. Suppose p is a prime and k ≥ 4 is an even integer. Then d_{k}(Γ_{0}(p)) > 0 unless

(p, k) ∈ {(2, 4), (2, 6), (2, 8), (2, 10),
(3, 4), (3, 6), (3, 8),
(5, 4), (5, 6), (7, 4), (11, 4)},

in which case d_{k}(Γ_{0}(p)) = 0.

Proof. From [Rib91], mod p eigenforms on Γ_{0}(p) of weight k arise exactly from mod p eigenforms on Γ_{0}(1) of weight (k/2)(p + 1). Moreover, there is an equality of dimensions of vector spaces:

dim S_{(k/2)(p+1)}(Γ_{0}(1)) + dim S_{(k/2)(p+1)−(p−1)}(Γ_{0}(1)) = dim S_{k}(Γ_{0}(p)).

Thus the dimension of S_{k}(Γ_{0}(p)) is bigger than the number of mod p eigenforms whenever dim S_{(k/2)(p+1)−(p−1)}(Γ_{0}(1)) is non-zero. The cases of dimension zero correspond exactly to the finite list of exceptions above, for which one can explicitly calculate that d_{k}(Γ_{0}(p)) = 0.

Note that for k = 2, however, there is a canonical identification of spaces

S_{(p+1)}(Γ_{0}(1), \overline{F}_{p}) ≃ S_{2}(Γ_{0}(p), \overline{F}_{p}),

described geometrically in [Gro90]. For k = 4, the data suggests that the discriminants d_{4}(Γ_{0}(p)) are significantly larger than zero for large p, and the table above suggests a formula of the form 2 · ⌊p/12⌋ (Not entirely coincidentally, this is the difference in dimension of the spaces S_{4}(Γ_{0}(p)) and S_{2(p+1)}(Γ_{0}(1))). This exact formula is not correct, however, as evidenced by the case when p = 139. If we consider the Hecke algebra T_{4} for p = 139 in more detail, however, we observe that T_{4} ⊗Q_{139} is ramified at 139, and in particular contains two copies of the field Q_{139}(\sqrt{139}). Just as in the case when k = 2 and p = 389, there is a “self congruence” between the associated ramified eigenforms and their Galois conjugates. For all other p in the range of the table, there is no ramification, and all congruences take place between distinct eigenforms. Such congruences are measured by the index of the Hecke algebra, which is defined to be the index of T in its normalisation T̃. If we are only interested in mod p congruences (rather than mod ℓ congruences for ℓ ≠ p), one can restrict to the index of T ⊗ Z_{p} inside its normalisation. There is a direct relation between the discriminant and the index. Suppose that T⊗Q_{p} = ∏ K_{i} for certain fields K_{i}/Q_{p} (We may assume here that T is not nilpotent, for otherwise both the discriminant and index are infinite). Then if i_{p}(Γ) = ord_{p}([T, T̃]), then

d_{p}(Γ) = 2i_{p}(Γ) + ∑ ord_{p}(Δ(K_{i}/Q_{p})).

If we now return to the example k = 4 and p = 139, we see that the discrepancy from the discriminant d_{p}(Γ_{0}(139)) = 24 to the estimate 2⌊139/12⌋ = 22 is exactly accounted for by the two eigenforms with coefficients in Q_{139}(\sqrt{139}), which contribute 2 to the above formula. This leads us to predict that the index is exactly given by the formula ⌊p/12⌋. Note that for primes p this is exactly the dimension of S_{p+3}(Γ_{0}(1)). Similar computations lead to the following more general conjecture.

Let k = 2m be an even integer and p a prime. Let T be the Hecke algebra associated to S_{k}(Γ_{0}(p)) and let T̃ be the integral closure of T in T ⊗ Q (which is a product of number fields).

Conjecture 1. Suppose p ≥ k − 1. Then

ord_{p}([T̃ : T]) = ⌊\frac{p}{12}⌋ · \binom{m}{2} + a(p, m),

where

a(p, m) = \begin{cases}0 & if p ≡ 1 (mod 12), \\ 3 · \binom{⌈\frac{m}{3}⌉}{2} & if p ≡ 5 (mod 12), \\ 2 · \binom{⌈\frac{m}{2}⌉}{2} & if p ≡ 7 (mod 12), \\ a(5, m) + a(7, m) & if p ≡ 11 (mod 12). \end{cases}

Here \binom{x}{y} is the binomial coefficient “x choose y”, and floor and ceiling are as usual. The conjecture is very false if k ≫ p.

When k = 2, the conjecture specializes to the assertion that [T̃ : T] is not divisible by p. A possibly more familiar concrete consequence of the conjecture is the following conjecture about elliptic curves. The modular degree of an elliptic curve E is the smallest degree of a surjective morphism X_{0}(N) → E, where N is the conductor of E.

Conjecture 2. Suppose E is an elliptic curve of prime conductor p. Then p does not divide the modular degree m_{E} of E.

Using the algorithm in [Wat02], M. Watkins has computed modular degrees of a huge number of elliptic curves of prime conductor p < 10^{7}, and not found a counterexample. Looking at smaller data, there is only one elliptic curve E of prime conductor p < 20000 such that the modular degree of E is even as big as the conductor of E, and that is a curve of conductor 13723. This curve has equation [1, 1, 1, −10481, 408636], modular degree m_{E} = 16176 = 2^{4} · 3 · 337. The modular degree can be divisible by large primes. For example, there is a Neumann-Setzer elliptic curve of prime conductor 90687593 whose modular degree is 1280092043, which is over 14 times as big as 90687593. In general, for an elliptic curve of conductor N, one has the estimate m_{E} ≫ N^{7/6−ϵ} (see [Wat04]).

5 Conjectures Inspired by Conjecture 1

First, some notation. Let p be an odd prime. Let Γ = Γ_{0}(p), and let

S_{k}(R) := S_{k}(Γ)^{new} ⊗ R.

The spaces S_{k} carry an action of the Hecke algebra T_{k}^{new}, and a Fricke involution w_{p}. If \frac{1}{2} ∈ R, the space S_{k} can be decomposed into + and − eigenspaces for w_{p}. We call the resulting spaces S_{k}^{+} and S_{k}^{−} respectively.

Similarly, let M_{k}^{+} and M_{k}^{−} be the +1 and −1 eigenspaces for w_{p} on the full spaces of new modular forms of weight k for Γ_{0}(p).

It follows from [AL70, Lem. 7] (which is an explicit formula for the trace to lower level) and the fact that U_{p} and w_{p} both preserve the new subspace, that the action of the Hecke operator U_{p} on S_{k} is given by the formula

U_{p} = −p^{(k−2)/2}w_{p}.

This gives rise to two quotients of the Hecke algebra:

T^{+} = T^{new}/(U_{p} + p^{(k−2)/2}) and T^{−} = T^{new}/(U_{p} − p^{(k−2)/2}).

where T^{+} and T^{−} act on S^{+} and S^{−}, respectively. Recall that T̃ is the normalization (integral closure) of T in T ⊗ Q. Let T̃^{new} denote the integral closure of T^{new} in T^{new} ⊗ Q.

Lemma 1. There are injections

T^{new} ↪ T^{+} ⊕ T^{−} ↪ T̃^{new}.

We now begin stating some conjectures regarding the rings T^{±}.

Conjecture 3. Let k < p − 1. Then T^{+} and T^{−} are integrally closed. Equivalently, all congruences between distinct eigenforms in S_{k}(\overline{Z}_{p}) take place between + and − eigenforms.

Note that for k = 2, there cannot be any congruences between + and − forms because this would force 1 ≡ −1 mod p, which is false, because p is odd. Thus we recover the conjecture that p ∤ [T̃ : T] when k = 2. Our further conjectures go on to describe explicitly the congruences between forms in S_{k}^{+} and S_{k}^{−}.

Let E_{2} be the non-holomorphic Eisenstein series of weight 2. The q-expansion of E_{2} is given explicitly by

E_{2} = 1 − 24 ∑_{n=1}^{∞} q^{n} (∑_{d|n} d) .

Moreover, the function E_{2}^{∗} = E_{2}(τ) − pE_{2}(pτ) is holomorphic of weight 2 and level Γ_{0}(p), and moreover on q-expansions, E_{2}^{∗} ≡ E_{2} mod p.

Lemma 2. Let p > 3. Let f ∈ M_{k}(Γ_{0}(p), \overline{F}_{p}) be a Hecke eigenform. Then θf is an eigenform inside S_{k+2}(Γ_{0}(p), \overline{F}_{p}).

Proof. One knows that ∂f = θf − kE_{2}f/12 is of weight k + 2. On qexpansions, E_{2} ≡ E_{2}^{∗} mod p, and thus for p > 3,

θf ≡ ∂f + kE_{2}^{∗}f/12 (mod p)

is the reduction of a weight k + 2 form of level p. It is easy to see that θf is a cuspidal Hecke eigenform.

Let us now assume Conjecture 3 and consider the implications for k = 4 in more detail. The space of modular forms M_{2}(Γ_{0}(p), \overline{F}_{p}) consists precisely of S_{2} and the Eisenstein series E_{2}^{∗}. The map θ defined above induces maps:

θ : S_{2}^{+}(\overline{F}_{p}) → S_{4}(\overline{F}_{p}), θ : M_{2}^{−}(\overline{F}_{p}) → S_{4}(\overline{F}_{p}).

The images are distinct, since θf = θg implies (with some care about a_{p}) that f = g.

Conjecture 4. Let f ∈ S_{2}(\overline{Z}_{p}) and g ∈ S_{4}(\overline{Z}_{p}) be two eigenforms such that θf ≡ g mod p. Then the eigenvalue of w_{p} on f and g have opposite signs.

Assuming this, we get inclusions:

θS_{2}^{+}(\overline{F}_{p}) ↪ S_{4}^{−}(\overline{F}_{p}), θM_{2}^{−}(\overline{F}_{p}) ↪ S_{4}^{+}(\overline{F}_{p}).

Now we are ready to state our main conjecture:

Conjecture 5. There is an Hecke equivariant exact sequence:

[figure: 0; θS_{2}^{+}(\overline{F}_{p}); S_{4}^{−}(\overline{F}_{p}); S_{4}^{+}(\overline{F}_{p}); θM_{2}^{−}(\overline{F}_{p}); 0.]

Moreover, the map S_{4}^{−}(\overline{F}_{p}) → S_{4}^{+}(\overline{F}_{p}) here is the largest such equivariant map between these spaces. Equivalently, a residual eigenform of weight 4 and level p occurs in both the + and − spaces if and only if it is not in the image of θ.

Let us give some consequences of our conjectures for the index of T^{new} inside its normalisation. Fix a residual representation \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(F_{q}) and consider the associated maximal ideal 𝔪 inside T_{4}. If \overline{ρ} lies in the image of θ then our conjecture implies that it is not congruent to any other eigenform. If \overline{ρ} is not in the image of θ, then it should arise exactly from a pair of eigenforms, one inside S_{4}^{+}(\overline{Q}_{p}) and one inside S_{4}^{−}(\overline{Q}_{p}). Suppose that q = p^{r}. If there is no ramification in T ⊗ Q over p (this is often true), then the + and − eigenforms will both be defined over the ring W (F_{q}) of Witt vectors of F_{q}. Since U_{p} = p on S_{4}^{−} and −p on S_{4}^{+}, these forms can be at most congruent modulo p. Thus the completed Hecke algebra (T_{4})_{𝔪} is exactly

{(a, b) ∈ W (F_{q}) ⊕ W (F_{q}), |a ≡ b mod p}.

One sees that this has index q = p^{r} inside its normalisation. Thus the (log of the) total index is equal to ∑ r_{i} over all eigenforms that occur inside S_{4}^{+} and S_{4}^{−}, which from our exact sequence we see is equal to

dim S_{4}^{−} − dim S_{2}^{+}.

Conjecture 1 when k = 4, would then follow from the equality of dimensions:

dim S_{4}^{−}(\overline{F}_{p}) − dim S_{2}^{+}(\overline{F}_{p}) = ⌊\frac{p}{12}⌋ .

We expect that something similar, but a little more complicated, should happen in general. In weight 2k, there are mod p^{k−r} congruences exactly between forms in the image of θ^{r−1} but not of θ^{r}.

5.1 Examples

We write small s’s and m’s for dimensions below.

Let p = 101. Then s_{2}^{+} = 1, m_{2}^{−} = 7 + 1 = 8, s_{4}^{−} = 9, s_{4}^{+} = 16. We predict the index should be 9 − 1 = 8 = ⌊101/12⌋. In the table below, we show the characteristic polynomials of T_{2} on S_{4}^{−} and S_{4}^{+}, and for weight 2, we take the characteristic polynomial of θT_{2} (or the same, taking F (x/2) where F (x) is the characteristic polynomial of T_{2}). Note that we have to add the Eisenstein series, which has characteristic polynomial x − 1 − 2, which becomes x − 6 ≡ x + 95 mod 101 under θ.

Factors of the Characteristic Polynomial of T_{2} for p = 101.

θS_{2}^{+}(\overline{F}_{101}) S_{4}^{−}(\overline{F}_{101})  S_{4}^{+}(\overline{F}_{101})  θM_{2}^{−}(\overline{F}_{101})
(x)  (x)  (x + 46)  (x + 95)
(x + 46)  (x + 95)  (x^{2} + 90x + 78)
(x^{2} + 58x + 100) (x^{2} + 58x + 100)  (x^{2} + 96x + 36)
(x^{5} + 2x^{4} + 27x^{3} (x^{2} + 90x + 78)  (x^{3} + 16x^{2}
+49x^{2} + 7x + 65) (x^{2} + 96x + 36)  +35x + 72)
(x^{3} + 16x^{2} + 35x + 72)
(x^{5} + 2x^{4} + 27x^{3}
+49x^{2} + 7x + 65)

Here are some further conjectures when k > 4.

Conjecture 6. Let p and k be such that 4 < k < p − 1. There is an Hecke equivariant exact sequence:

[figure: 0; θS_{k−2}^{+}(\overline{F}_{p}); S_{k}^{−}(\overline{F}_{p}); S_{k}^{+}(\overline{F}_{p}); θS_{k−2}^{−}(\overline{F}_{p}); 0.]

Moreover, all forms not in the image of θ contribute maximally to the index (a factor of p^{(k−2)/2}). Thus the total index should be equal to

\frac{(k − 2)}{2}(dim S_{k}^{+} − dim S_{k−2}^{−}) + the index at level p and weight k − 2.

This is the sum

∑_{n=2}^{k} \frac{(2n − 2)}{2}(s_{2n}^{+} − s_{2n−2}^{−}).

When k = 4, we need to add the Eisenstein series to S_{2}^{−} in our previous conjecture. Note that s_{k}^{+} − s_{k−2}^{−} = s_{k}^{−} − s_{k−2}^{+} for k > 4 (and with s_{2}^{−} replaced by m_{2}^{−} when k = 2). This follows from our conjectures, but can easily be proved directly. As an example, when p = 101, we have s_{2}^{+} = 1, s_{4}^{−} = 9, s_{6}^{+} = 17, s_{8}^{−} = 26, s_{10}^{+} = 34, s_{12}^{−} = 42, s_{14}^{+} = 51, and so we would predict the indexes I_{k} to be as given in the following table:

\begin{matrix}k & I_{k}? \\ 2 & 0 \\ 4 & 8 = 8 + 0 \\ 6 & 24 = 24 + 0 \\ 8 & 51 = 48 + 3 \\ 10 & 83 = 80 + 3 \\ 12 & 123 = 120 + 3 \\ 14 & 177 = 168 + 9 \end{matrix}

This agrees with our conjectural formula, which says that the index should be equal in this case to

8\binom{k/2}{2} + 3\binom{⌈k/6⌉}{2}.

it also agrees with computation.

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