Discriminants of Hecke Algebras at Prime Level

William A. Stein^{∗}

September 24, 2002

1 Introduction

In this paper, we study p-divisibility of discriminant 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 conjecture about indexes of Hecke algebras in their normalization which implies (if true) the surprising conjecture that there are no mod p congruences between non-conjugate newforms in S_{2}(Γ_{0}(p)).

2 Discriminants of Hecke Algebras

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

Definition 2.1 (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})), which is well defined modulo squares of units in A.

When R = ℤ the discriminant is well defined, since the only units are ±1.

Proposition 2.2. Suppose R is a field. Then A has discriminant 0 if and only if A is separable over R, i.e., for every extension R′ of R, the ring A ⊗ R′ contains no nilpotents.

The following proof is summarized from [Mat86, §26]. If A contains a nilpotent then that nilpotent is in the kernel of the trace pairing. 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.

2.1 The Discriminant Valuation

Let Γ be a congruence subgroup of SL_{2}(ℤ), e.g., Γ = Γ_{0}(p) or Γ_{1}(p). For any integer k ≥ 1, let S_{k}(Γ) denote the space of holomorphic weight-k cusp forms for Γ. Let

𝕋 = ℤ[…, T_{n}, …] ⊂ End(S_{k}(Γ))

∗This will evolve into a joint paper with Frank Calegari.

be the associated Hecke algebra. Then 𝕋 is a commutative ring that is free of finite rank as a ℤ-module. Also of interest is the image 𝕋^{new} of 𝕋 in End(S_{k}(Γ)^{new}). Example 2.3. Let Γ = Γ_{0}(243). Since 243 = 3^{5}, there are nilpotents in 𝕋, so disc(𝕋) = 0. A computation shows that

disc(𝕋^{new}) = 2^{13} · 3^{40}.

Definition 2.4 (Discriminant Valuation). Let p be a prime and suppose that Γ = Γ_{0}(p) or Γ_{1}(p). The discriminant valuation is

d_{k}(Γ) = ord_{p}(disc(𝕋)).

3 Motivation and Applications

Let p be a prime and suppose that Γ = Γ_{0}(p) or Γ_{1}(p). A normalized 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, normalized so that a_{1} = 1. The quantity d_{k}(Γ) is of interest because it measures mod p congruences between normalized eigenforms in S_{k}(Γ).

Proposition 3.1. Suppose that d_{k}(Γ) is finite. Then the discriminant valuation d_{k}(Γ) is nonzero if and only if there is a mod-p congruence between two Hecke eigenforms in S_{k}(Γ) (note that the two congruent eigenforms might be Galois conjugate).

Proof. It follows from Proposition 2.2 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). Using the standard bijection between homomorphisms and normalized eigenforms, we see that 𝕋 ⊗ \overline{𝔽}_{p} is not separable if and only if there is a mod-p congruence between two normalized eigenforms.

Example 3.2. 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, so T_{2} generated 𝕋 ⊗ ℤ_{389} as a ring. (If it generated a subring of 𝕋 ⊗ ℤ_{389} of finite index, then the discriminant of f would be divisible by 389^{2}.)

Modulo 389 the 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 ⊗ \overline{𝔽}_{389} is not separable since the image of T_{2} + 175 is nilpotent (its square is 0). There are 32 eigenforms over ℚ_{2} but only 31 mod-389 eigenforms, so there must be a congruence. Let F be the 389-adic newform whose a_{2} term is a root of

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

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

Example 3.3. The discriminant of the Hecke algebra 𝕋 associated to S_{2}(Γ_{0}(389)) is

2^{53}·3^{4}·5^{6}·31^{2}·37·389·3881·215517113148241·477439237737571441

I computed this using the following algorithm, which was suggested by Hendrik Lenstra. Using the Sturm bound I found a b such that T_{1}, …, T_{b} generate 𝕋 as a ℤ-module. I then found a subset B of the T_{i} that form a ℚ-basis for 𝕋 ⊗_{ℤ} ℚ. Next, viewing 𝕋 as a ring of matrices acting on ℚ^{32}, I found a random vector v ∈ ℚ^{32} such that the set of vectors C = {T (v) : T ∈ B} is linearly independent. Then I wrote each of T_{1}(v), …, T_{b}(v) as ℚ-linear combinations of the elements of C. Next I found a ℤ-basis D for the ℤ-span of these ℚ-linear combinations of elements of C. Tracing everything back, I find the trace pairing on the elements of D, and deduce the discriminant by computing the determinant of the trace pairing matrix. The most difficult step is computing D from T_{1}(v), …, T_{b}(v) expressed in terms of C, and this explains why we embed 𝕋 in ℚ^{32} instead of viewing the elements of 𝕋 as vectors in ℚ^{32^{2}}. This whole computation takes one second on an Athlon 2000 processor.

4 Data About Discriminant Valuations

4.1 Weight Two

Theorem 4.1. The only prime p < 60000 such that d_{2}(Γ_{0}(p)) > 0 is p = 389. (Except possibly 50923 and 51437, which I haven’t finished checking yet.)

For various reasons this theorem has been referred to in [Rib99, MS01, OW02, MO02]. For example, it is used in [MS01] in bounding the field of definition of certain torsion points on elliptic curves.

Proof. This is the result of a large computer computation, and perhaps couldn’t be verified any other way, since I know of no general theorems about d_{2}(Γ_{0}(p)). The rest of this proof describes how I did the computation, so you can be convinced that there is valid mathematics behind my computation, and that you could verify the theorem given sufficient time. The computation described below took about one week using 10 Athlon 2000MP processors. In 1999 I had checked the result stated above but only for p < 14000 using a completely different implementation of the algorithm and a 200Mhz Pentium computer. These 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 algegra 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).

Most levels p < 60000 were ruled out by computing characteristic polynomials of Hecke operators using an algorithm that David Kohel and the author implemented in MAGMA ([?]), which is based on the Mestre-Oesterle method of graphs (our implementation is “The Modular of Supersingular Points” package that comes with MAGMA). The author computed disc(T_{q}) modulo p for several primes q, and in most cases found a q such that this discriminant is nonzero. The following table summarizes 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 the trick the author 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. Though computing T_{n} on the full space of modular symbols is quite hard, it turns out that there is an algorithm that quickly computes T_{n} on subspaces of modular symbols with small dimension (see [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_{1} has distinct roots, and they are n = 34, 47, 53, and 89. We then computd charpoly(T_{34}) directly on the whole space and found that it has distinct roots modulo p.

4.2 Higher Weight Data

1. 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.

\begin{matrix}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 \end{matrix}

5 Conjectures

Conjecture 5.1. 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.

Conjecture 5.2. Suppose p > 2 is a prime and k ≥ 3 is an integer. If

(p, k) ∉ {(3, 3), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8),
(5, 3), (5, 4), (5, 5), (5, 6), (5, 7)
(7, 3), (7, 4), (7, 5), (11, 3), (11, 4), (11, 5),
(13, 3), (17, 3), (19, 3)}

then d_{k}(Γ_{1}(p)) > 0.

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 normalization of 𝕋̃ in 𝕋 ⊗ ℚ. The following conjecture is shocking. It is amazing something so simple might be true.

Conjecture 5.3.

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}

In particular, when k = 2 we conjecture that [𝕋̃ : 𝕋] is not divisible by p.

Here \binom{x}{y} is the binomial coefficient “x choose y”, and floor and ceiling are as usual. We have checked this conjecture against significant numerical data.

References

[Mat86] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986, Translated from the Japanese by M. Reid. MR 88h:13001

[MO02] Momose and Ozawa, Rational points of modular curves X_{split}(p), Preprint (2002).

[MS01] L. Merel and W. A. Stein, The field generated by the points of small prime order on an elliptic curve, Internat. Math. Res. Notices (2001), no. 20, 1075–1082. MR 1 857 596

[OW02] K. Ono and McGraw W., Modular form congruences and selmer groups, Preprint (2002).

[Rib99] K. A. Ribet, Torsion points on J_{0}(N) and Galois representations, Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin, 1999, pp. 145–166. MR 2001b:11054

[Ste00] W. A. Stein, Explicit approaches to modular abelian varieties, Ph.D. thesis, University of California, Berkeley (2000).