Modular Degrees of Elliptic Curve

and

Discriminants of Hecke Algebras

William Stein^{∗}

           http://modular.fas.harvard.edu

ANTS VI, June 18, 2004

∗Joint with F. Calegari

Goal

Let p be a prime. The goal of this talk is to explain and justify the following increasingly general Calegari-Stein conjectures:

Conjecture 1. (–). If E/Q is an elliptic curve of conductor p, then the modular degree m_{E} of E is not divisible by p.

Conjecture 2. (–). If T_{2}(p) is the Hecke algebra associated to S_{2}(Γ_{0}(p)), then p does not divide the index of T_{2}(p) in its normalization.

Conjecture 3. (–). If p > k −1, then there is an explicit formula for the p-part of the index of T_{k}(p) in its normalization.

Conj 1: If E of conductor p, then p ∤ m_{E}.

Vandiver: Conjecture 1 looks like Vandiver’s conjecture, which asserts that p ∤ h_{p}^{−}. (Note Flach’s Selmer group connection.)

Data: (Watkins) For p < 10^{7} there are 52878 curves in Stein-Watkins table. No counterexamples to conjecture 1. There are 23 curves such that m_{E} is divisible by a prime ℓ > p. For example the curve y^{2} + xy = x^{3} − x^{2} − 391648x − 94241311 of prime conductor p = 4847093 has modular degree 2 · 21695761. Smallest p with ℓ > p is p = 1194923.

Ratio: Max ratio m_{E}/p is ∼ 23.2, attained for p = 7944197. First curve with m_{E}/p > 1 has level 13723, where m_{E} = 16176 = 2^{4} · 3 · 337. Smallest m_{E}/p > 1 is p = 1757963; m_{E} = p + 715.

Conjecture is consistent with ABC-conjecture (m_{E} is O(p^{2+ε} )).

Cuspidal Modular Forms

Congruence Subgroup:

Γ_{0}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z) such that N | c\} .

Cusp Forms: S_{k}(N) = {f : 𝔥 → C such that

f(γ(z)) = (cz + d)^{−k}f(z) all γ ∈ Γ_{0}(N),

and f is holomorphic at the cusps  }

Fourier Expansion:

f = _{n≥1}^{∑} a_{n}e^{2πizn} = _{n≥1}^{∑} a_{n}q^{n} ∈ C[[q]].

Modular Forms Examples

S_{k}(N) = 0 if k is odd, so we will not consider odd k further.

For k ≥ 2, a basis of S_{k}(N) can be computed to any given precision using modular symbols (e.g., my MAGMA packages). Appears that no formal analysis of complexity has been done. Certainly polynomial time in N and required precision.

MAGMA CODE
> S := CuspForms(37,2);
> Basis(S);
[
    q + q^3 - 2*q^4 - q^7 + O(q^8),
    q^2 + 2*q^3 - 2*q^4 + q^5 - 3*q^6 + O(q^8)
]

Basis for S_{14} (11):

> S := CuspForms(11,14); SetPrecision(S,17);
> Basis(S);
q   - 74*q^13 - 38*q^14 + 441*q^15 + 140*q^16 + O(q^17),
q^2 - 2*q^13 + 78*q^14 + 24*q^15 - 338*q^16 + O(q^17),
q^3 + 18*q^13 - 72*q^14 + 89*q^15 + 492*q^16 + O(q^17),
q^4 + 12*q^13 + 31*q^14 - 18*q^15 - 193*q^16 + O(q^17),
q^5 - 10*q^13 + 46*q^14 - 63*q^15 - 52*q^16 + O(q^17),
q^6 + 11*q^13 - 18*q^14 - 74*q^15 - 4*q^16 + O(q^17),
q^7 - 7*q^13 - 16*q^14 + 42*q^15 - 84*q^16 + O(q^17),
q^8 - q^13 - 16*q^14 - 18*q^15 - 34*q^16 + O(q^17),
q^9 - 8*q^13 - 2*q^14 - 3*q^15 + 16*q^16 + O(q^17),
q^10 - 5*q^13 - 2*q^14 - 6*q^15 + 14*q^16 + O(q^17),
q^11 + 12*q^13 + 12*q^14 + 12*q^15 + 12*q^16 + O(q^17),
q^12 - 2*q^13 - q^14 + 2*q^15 + q^16 + O(q^17)

Hecke algebras

Hecke Operators: Let p be a prime.

T_{p} (_{n≥1}^{∑} a_{n} · q^{n}) = _{n≥1}^{∑} a_{nr} · q^{n} + p^{k−1} _{n≥1}^{∑} a_{n} · q^{np}

(If p | N, drop the second summand.) This preserves S_{k}(N), so defines a linear map

T_{p} : S_{k}(N) → S_{k}(N).

Similar definition of T_{n} for any integer n. Hecke Algebra: A commutative ring:

T_{k}(N) = Z[T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, …] ⊂ End_{C}(S_{k}(N))

Computing Hecke Algebra

Fact: T_{k}(N) = Z[T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, …] is free as a Z-module of rank equal to dim S_{k}(N).

Sturm Bound: T_{k}(N) is generated as a Z-module by T_{1}, T_{2}, …, T_{b}, where b is the ceiling of

\frac{k}{12} · N · _{p|N}^{∏} (1 − \frac{1}{p}) .

Example: For N = 37, bound is 7, and T_{2}(37) has Z-basis

T_{1} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) and T_{2} = (\begin{matrix}−2 & 1 \\ 0 & 0 \end{matrix}).

There are several other T_{k}(N)-modules isomorphic to S_{2}(N), and I use these instead to compute T_{k}(N) as a ring.

Discriminants

The discriminant of T_{k}(N) is an integer. It measures ramification, or what’s the same, congruences between simultaneous eigenvectors for T_{k}(N), hence is related to the modular degree. Discriminant:

Disc (T_{k}(N)) = Det (Tr (t_{i} · t_{j})),

where t_{1}, …, t_{n} are a basis for T_{k}(N) as a free Z-module. Examples:

Disc (T_{2}(37)) = Det (\begin{matrix}2 & −2 \\ −2 & 4 \end{matrix}) = 4

Disc (T14 (11)) = 2^{46} · 3^{14} · 5^{2} · 11^{42} · 79 · 241 · 1163 · 40163 · 901181111 · 47552569849·124180041087631·205629726345973.

Ribet’s Question

I became interested in computing with modular forms when I was a grad student and Ken Ribet started asking: Question: (Ribet, 1997) Is there a prime p so that p | Disc (T_{2}(p))? Ribet had proved a theorem about X_{0}(p) ∩ J_{0}(p)_{tor} under the hypothesis that p ∤ T_{2}(p), and wanted to know how restrictive his hypothesis was. Note that when k > 2, usually p | Disc (T_{k}(p)). Using a PARI script of Joe Wetherell, I set up a computation on my laptop and found exactly one example: p = 389.

Index in the Normalization

Last year I checked that for p < 50000 there are no other examples in which p | Disc (T_{2}(p)). For this I used the Mestre method of graphs, which involves computing with the free abelian group on the supersingular j-invariants in F_{p^{2}} of elliptic curves.

Let T̃_{k}(p) be the normalization of T_{k}(p). Since T_{k}(p) is an order in a product of number fields, T̃_{k}(p) is the product of the rings of integers of those number fields.

It turned out that Ribet could prove his theorem under the weaker hypothesis that p ∤ [T̃_{k}(p) : T_{k}(p)]. I was unable to find a counterexample to this divisibility. (Note: Matt Baker’s Ph.D. was a proof of the full theorem using different methods.)

Conjecture 2

Conjecture 2. (–). If T_{2}(p) is the Hecke algebra associated to S_{2}(Γ_{0}(p)), then p does not divide the index of T_{2}(p) in its normalization.

The primes that divide [T̃_{k}(p) : T_{k}(p)] are called congruence primes. They are the primes of congruence between non-Gal(\overline{Q}/Q)- conjugate eigenvectors for T_{k}(p). Using this observation and another theorem of Ribet (and Wiles et al. modularity), one see that a “no” answer to the above question implies that p does not divide the modular degree of any elliptic curve of conductor p. This is why Conjecture 2 implies Conjecture 1.

But is there any reason to believe Conjecture 2, beyond knowing that it is true for p < 50000?

Higher Weight

Recall that

Disc (T14 (11)) = 2^{46} · 3^{14} · 5^{2} · 11^{42} · 79 · 241 · 1163 · 40163 · 901181111 ·

47552569849·124180041087631·205629726345973.

Notice the large power of 11. Upon computing the p-maximal order in T_{14} (11) ⊗_{Z} Q, we find that 11 ∤ Disc (T̃_{14} (11)), so all the 11 is in the index of T_{14} (11) in T̃_{14} (11). Thus

ord_{11} ([T̃_{14} (11) : T_{14} (11)]) = 21.

Data for k = 4

Each row contains p and ord_{p}(Disc (T_{4}(17))). E.g., ord_{17} (Disc (T_{4}(17))) = 2.

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

F. Calegari (during a talk I gave): Except for 389, there is clearly a pattern!! Calegari and I computed 2 · [T̃_{4}(p) : T_{4}(p)] and obtained the same numbers as above, except for p = 389 which now gives 64. We also considered many examples where

2 · [T̃_{4}(p) : T_{4}(p)] ≠ Disc (T_{k}(p)).

Conjecture 3

In all cases, we found the following amazing pattern: Conjecture 3. Suppose p ≥ k − 1. Then

ord_{p}([T̃_{k}(p) : T_{k}(p)]) = ⌊\frac{p}{12} ⌋ · \binom{k/2}{2} + a(p, k),

where

a(p, k) = \begin{cases}0 & if p ≡ 1 (mod 12), \\ 3 · \binom{⌈\frac{k}{6}⌉}{2} & if p ≡ 5 (mod 12), \\ 2 · \binom{⌈\frac{k}{4}⌉}{2} & if p ≡ 7 (mod 12), \\ a(5, k) + a(7, k) & if p ≡ 11 (mod 12). \end{cases}

Warning: The conjecture is false without the constraint that p be big compared to k. Though it works for our running example p = 11, k = 14, where the formula yields 0 + 3 · \binom{3}{2} + 2 · \binom{4}{2} = 9 + 12 = 21, which is correct.

Summary

For a long time I had no idea whether to conjecture that there should or shouldn’t be mod p congruence between nonconjugate eigenforms, or equivalently, whether p divides modular degrees at prime level. By considering higher weight and computing, a simple conjectural formula emerges, which when specialized to 2 is the conjecture that there are no mod p congruences.

Future Direction. Explain why there are so many mod p congruences at level p, when k ≥ 4. See paper for a strategy.

Computational Question. Push computation of ord_{p}(Disc (T_{2}(p))) to 100000 using Wiedemann’s minimal polynomial algorithm.

Vandiver-ish Question. Investigate the connection between Conjecture 1 and Flach’s results on modular degrees annihilating Selmer groups.