FACTORING HECKE

POLYNOMIALS MODULO A PRIME

J.B. Conrey D.W. Farmer P.J. Wallace

Abstract. Let T_{p,k}^{N,χ}(x) be the characteristic polynomial of the Hecke operator T_{p} acting on the space of cusp forms S_{k}(N, χ). We describe the factorization of T_{p,k}^{N,χ}(x) mod ℓ as k varies, and we explicitly calculate those factorizations for N = 1 and small ℓ. These factorizations are used to deduce the irreducibility of certain T_{q,k}^{1,1}(x) from the irreducibility of T_{2,k}^{1,1}(x).

1. Introduction and statement of results

Let S_{k}(N, χ) be the space of holomorphic cusp forms of integral weight k for the Hecke congruence subgroup Γ_{0}(N), and denote by T_{p,k}^{N,χ}(x) the characteristic polynomial of the Hecke operator T_{p} acting on S_{k}(N, χ). Let S_{k}(N) = S_{k}(N, 1) and set T_{p,k} = T_{p,k}^{1,1}. A conjecture of Maeda asserts that T_{p,k}(x) is irreducible and has full Galois group over ℚ. This conjecture is related to the nonvanishing of L–functions [KZ][CF], and to constructing base changes to totally real number fields for level 1 eigenforms [HM]. Maeda’s conjecture has been checked for all primes p < 2000 and weights k ≤ 2000 [B][CF][FJ].

The methods which have been used to check Maeda’s conjecture involve computing the factorization of T_{2,k}(x) mod ℓ for various ℓ. One searches for enough factorizations to deduce that T_{2,k}(x) is irreducible and has full Galois group over ℚ. With a small amount of additional calculation it is possible to deduce the same conclusion for other T_{p,k}(x). The method of translating information about T_{2,k}(x) to information about T_{p,k}(x) is related to the following result of James and Ono.

Theorem[JO]. Suppose q and ℓ are distinct primes. Then

# {prime p < X | T_{p,k}^{N,χ}(x) ≡ T_{q,k}^{N,χ}(x) mod ℓ} ≫ X/ log X.

In particular, if T_{p,k}^{N,χ}(x) is irreducible mod ℓ for some p, then the same holds for a positive proportion of primes p.

Typeset by AMS-TEX

Research of the first two authors supported by the American Institute of Mathematics. Research of the first author supported in part by a grant from the NSF. A portion of this work resulted from the third author’s Honors Thesis at Bucknell University.

The above result makes use of the Chebotarev density theorem and deep results of Deligne, Serre, and Shimura on the Galois representations associated to modular forms. In this paper we describe how to determine, with a finite calculation, the factorization of T_{p,k}^{N,χ}(x) mod ℓ for any k. We use those factorizations to give an easy proof of the following result, which is similar to the second statement in the theorem given above.

Theorem 1. If T_{n,k}(x) is irreducible and has full Galois group for some n, then T_{p,k}(x) is irreducible and has full Galois group for p prime and p ≢ ±1 mod 5, or p ≢ ±1 mod 7.

Note that the conclusion of Theorem 1 holds for 5/6 of all primes p. Farmer and James [FJ] have used a version of the above result to show that T_{p,k}(x) is irreducible and has full Galois group if p < 2000 and k ≤ 2000.

In section 2 we describe the factorizations of T_{p,k}^{N,χ}(x) mod ℓ. In section 3 we give some examples. In section 4 we deduce some corollaries and prove Theorem 1.

Throughout the paper, p, q, and ℓ are distinct rational primes; N and n are positive rational integers with N, np, q, and ℓ pairwise coprime; k is a positive rational integer, and χ is a character of conductor N with χ(−1) = (−1)^{k}. We write T_{n,k}^{N,χ}(x) for the characteristic polynomial of T_{n} acting on the space of cusp forms S_{k}(N, χ) and put T_{n,k}(x) = T_{n,k}^{1,1}(x). When we refer to the irreducibility or to the Galois group of a polynomial, we will mean over ℚ. For prime p we write p^{A}||b to mean p^{A}|b and p^{A+1} ∤ b.

The authors thank J–P. Serre for helpful comments concerning the calculations in this paper.

2. Patterns in the factorization of T_{p,k}^{N,χ}(x) mod ℓ

Fix p, ℓ, N, and χ. In this section we describe the factorization of T_{p,k}^{N,χ}(x) mod ℓ as k varies. We show that those factorizations follow a pattern. In principle one can use this pattern to determine T_{p,k}^{N,χ}(x) mod ℓ for any k, after an initial calculation. In the following section we give some examples for N = 1 and small ℓ.

The idea here is to consider S_{k}(N, χ) mod ℓ and T_{n} mod ℓ, and to find a basis B_{k}(N, χ) of S_{k}(N, χ) with the following nice property.

Lemma 1. Let [T_{n}]_{k,N,χ} be the matrix of T_{n} with respect to the basis B_{k}(N, χ). Then [T_{n}]_{k,N,χ} is block upper–triangular and [T_{n}]_{k,N,χ} ⊂ [T_{n}]_{k+ℓ−1,N,χ}, where the smaller matrix is a block in the upper left corner of the larger matrix. In particular, T_{n,k+ℓ−1}^{N,χ}(x) ≡ g(x) T_{n,k}^{N,χ} mod ℓ for some polynomial g(x).

Proof. We define the basis B_{k}(N, χ) as follows. The classical congruence for the level 1 Eisenstein series E_{ℓ−1} ≡ 1 mod ℓ gives an inclusion S_{k}(N, χ) ⊂ S_{k+ℓ−1}(N, χ) mod ℓ, given by multiplication by E_{ℓ−1}. Choose B_{k}(N, χ) to respect this inclusion. The Hecke operator T_{n} mod ℓ also respects this inclusion because we have

(T_{n}f) (z) = \frac{1}{n} ∑_{ad=n} χ(a)a^{k} ∑_{0≤b<d} f (\frac{az + b}{d})

and a^{k} ≡ a^{k+ℓ−1} mod ℓ. This proves Lemma 1.

By Lemma 1, there is a sequence of polynomials f_{j} ∈ (ℤ/ℓℤ)[x], depending only on N, χ, p, ℓ, and (k mod ℓ − 1), so that

T_{p,k}^{N,χ}(x) ≡ ∏_{j=1}^{J} f_{j}(x) mod ℓ.  (2.1)

The following Proposition explains why it is easy to understand T_{p,k}^{N,χ}(x) mod ℓ for any k.

Proposition 1. The sequence f_{j} in (2.1) is periodic.

Thus, the calculation of T_{p,k}^{N,χ}(x) mod ℓ for any k reduces to a finite calculation. The periodicity of f_{j} follows from the isomorphism W_{k+ℓ+1} = W_{k}[1]. Here W_{k} = M̃_{k+ℓ−1}/M̃_{k}, with M̃_{k} the 𝔽_{ℓ} vector space obtained by reducing M_{k} mod ℓ. See [J]. We give a different proof based on the trace formula.

Proof. The degree of each f_{j} is bounded by

M = max_{k} (dim S_{k+ℓ−1}(N, χ) − dim S_{k}(N, χ)) .

This is finite because dim S_{k}(N, χ) grows linearly as a function of k.

The polynomial f_{j} has at most M roots, and the coefficients of f_{j} are symmetric functions in those roots. These can be expressed as polynomials in the traces of T_{p}, T_{p}^{2},..., T_{p}^{M}, which in turn can be expressed as polynomials in T_{p^{j}} with j ≤ M. Thus, the coefficients of f_{j} are just polynomials in the traces of T_{p}, T_{p^{2}},..., T_{p^{M}} mod ℓ.

Let σ_{k}^{N,χ}(T_{n}) denote the trace of T_{n} acting on S_{k}(N, χ). We need only show that σ_{k}^{N,χ}(T_{n}) mod ℓ is periodic as a function of k. This follows from the Eichler–Selberg trace formula. We state the trace formula in the following way, retaining only the features necessary for our proof. For details see [SV] or [C].

Lemma 2 (The trace formula). There are explicit algebraic numbers A, B_{m}, and C_{d}, depending only on N, n, and χ, such that

σ_{k}^{N,χ}(T_{n}) = A n^{k/2}(k − 1)χ(\sqrt{n}) + ∑_{|m|≤2\sqrt{n}} B_{m} \frac{η_{m}^{k−1} − η̄_{m}^{k−1}}{η_{m} − η̄_{m}} + ∑_{{d|n \atop 0<d≤\sqrt{n}}} C_{d} d^{k},

where η_{m} = \frac{m + \sqrt{m^{2} − 4n}}{2}. Furthermore, each of A, B_{m}, and C_{d} is integral modulo the rational prime ℓ for ℓ ≥ 5 and ℓ ∤ nN.

Each term in Lemma 2 is periodic mod ℓ as a function of k; therefore so is σ_{k}^{N,χ}(T_{n}). One slight complication is that if n is a square mod ℓ and ℓ < 4n, then for some m we may have η_{m} − η̄_{m} ≡ 0 mod ℓ^{K} for K ≥ 1. In this case choose L so that η_{m}^{k+L} ≡ η_{m}^{k} mod ℓ^{K+1}. This proves Proposition 1.

In many cases the period of f_{j} is too large to actually compute on available computers. However, for N = 1 and ℓ reasonably small, the computation is tractable. In the next section we give several examples, and in the following section we use those examples to deduce Theorem 1.

3. Some factorizations of level 1 Hecke polynomials

We give some examples of the factorizations of Hecke polynomials mod ℓ described in the previous section. The easiest examples to calculate are for N = 1 and ℓ ≤ 7 or ℓ = 13. In those cases

dim S_{k+ℓ−1}(1, 1) ≤ 1 + dim S_{k}(1, 1),

so T_{p,k}(x) mod ℓ factors completely and we only need the trace of T_{p} to determine each factor.

There is a further simplification for ℓ ≤ 7 arising from the classification of cusp forms mod ℓ. For normalized Hecke eigenforms f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n}, write f ≡ g mod ℓ to mean a_{n} ≡ b_{n} mod ℓ for (n, ℓ) = 1. For any particular ℓ there are only finitely many congruence classes of Hecke eigenforms mod ℓ. These are given explicitly by Serre [Ser] for ℓ ≤ 23.

For ℓ = 2 we have a_{p} ≡ 0 mod 2, so T_{p,k}(x) ≡ x^{d_{k}} mod 2 for all p ≠ 2. For ℓ = 3, 5, or 7, we have a_{p} ≡ p^{m} + p^{n} mod ℓ for p ≠ ℓ, where m and n depend only on ℓ and the Hecke eigenform. Thus, if p ≡ q mod ℓ then a_{p} ≡ a_{q} mod ℓ, so T_{p,k}(x) ≡ T_{q,k}(x) mod ℓ if p ≡ q mod ℓ. For ℓ ≤ 7 this reduces the determination of T_{p,k}(x) mod ℓ for all p and k to the calculation of a few cases. The results of those calculations are presented in the following Theorem.

Theorem 2a. For prime ℓ ≤ 7 and p ≠ ℓ, the Hecke polynomial T_{p,k}(x) factors as

T_{p,k}(x) ≡ ∏_{j=1}^{d_{k}}(x − a_{j}) mod ℓ

where {a_{j}} is a periodic sequence depending only on p mod ℓ and k mod (ℓ − 1). We have

T_{p,k}(x) ≡ \{ \begin{matrix}(x − 2)^{d_{k}} mod 3, & if p ≡ 1 mod 3 \\ x^{d_{k}} mod 3 & if p ≡ 2 mod 3 \end{matrix}

For ℓ = 5 or 7 the results are summarized in the tables below. The rows are labeled by the smallest prime in each congruence class mod ℓ, the columns are labeled by the congruence class of k mod (ℓ−1), and the table entry gives one period of the sequence {a_{j}}. The tables are sufficient to determine T_{p,k}(x) mod ℓ for all p because if ℓ ≤ 7 and p ≡ q mod ℓ then T_{p,k}(x) ≡ T_{q,k}(x) mod ℓ.

\begin{matrix}ℓ = 5 & 0 & 2 \\ p = 11 & (2) & (2) \\ 2 & (1,4) & (2,3) \\ 3 & (2,3) & (1,4) \\ 19 & (0) & (0) \end{matrix}  \begin{matrix}ℓ = 7 & 0 & 2 & 4 \\ p = 29 & (2) & (2) & (2) \\ 2 & (4,5) & (1,3) & (6,2) \\ 3 & (0,1,0,6) & (0,3,0,4) & (5,0,2,0) \\ 11 & (1,3) & (4,5) & (6,2) \\ 5 & (0,3,0,4) & (0,1,0,6) & (2,0,5,0) \\ 13 & (0) & (0) & (0) \end{matrix}

For ℓ > 7 there are no simple relationships between T_{p,k} and T_{q,k} mod ℓ, so each T_{p,k} mod ℓ must be calculated separately. We give an example mod 13.

Theorem 2b. We have a factorization T_{2,k}(x) ≡ ∏_{j=1}^{d_{k}}(x − a_{j}) mod 13, where {a_{j}} is a sequence of period 14 depending only on k mod 12. The first 14 terms of each sequence are given in the following table, where each row corresponds to a congruence class of k mod 12.

\begin{matrix}ℓ = 13 & (a_{1} … a_{14}) \\ k ≡ 0 mod 12 & (2, 12, 9, 4, 1, 11, 5, 11, 1, 4, 9, 12, 2, 8) \\ 2 & (4, 11, 5, 8, 2, 9, 10, 9, 2, 8, 5, 11, 4, 3) \\ 4 & (8, 6, 8, 9, 10, 3, 4, 5, 7, 5, 4, 3, 10, 9) \\ 6 & (5, 3, 12, 3, 5, 7, 6, 8, 10, 1, 10, 8, 6, 7) \\ 8 & (1, 10, 6, 11, 6, 10, 1, 12, 3, 7, 2, 7, 3, 12) \\ 10 & (11, 2, 7, 12, 9, 12, 7, 2, 11, 6, 1, 4, 1, 6) \end{matrix}

The calculations for Theorem 2 were done in Mathematica. The method was to find a basis for S_{k}(1) using Δ, E_{4}, and E_{6}, and then explicitly compute the action of the Hecke operator. Representative cases were checked using the trace formula.

The period of a_{j} given in Theorem 2 is shorter than might have been predicted from Lemma 2. If p is not a square mod ℓ then (m^{2} − 4p, ℓ) = 1 for all m, so Fermat’s little theorem gives η_{m}^{k} ≡ η_{m}^{k+ℓ^{2}−1} mod ℓ, which implies σ_{k}(T_{p}) = σ_{k+ℓ^{2}−1}(T_{p}) mod ℓ. This implies that the a_{j} mod ℓ has period at most (ℓ^{2} − 1)/12. This is the actual period given for those cases in Theorem 2a and Theorem 2b. If p is a square mod ℓ then the period of σ_{k}(T_{p}) is much larger, leading to a larger upper–bound for the period of a_{j}. (The largest case we needed is p = 11, m = 3, ℓ = 7, where we have η_{3}^{295} ≡ η_{3} mod 7^{2}.) However, as can be seen in the examples, the period of a_{j} is actually smaller. J–P. Serre has pointed out to us that the elementary result W_{k+p+1} = W_{k}[1] gives the indicated periodicity, and this can also be used to explain the various patterns which appear in the table in Theorem 2b.

4. Proof of Theorem 1

In this section we deduce some consequences of the factorizations given in Theorem 2 and we use those factorizations to prove Theorem 1.

Corollary. Suppose for some n that T_{n,k}(x) is irreducible over ℚ. Then T_{p,k}(x) is irreducible if either of the following holds:

i) dim S_{k}(1) is odd and p ≢ ±1 mod 5 or p ≢ ±1 mod 7, or ii) dim S_{k}(1) ≡ 2 mod 4 and p ≡ 3 or 5 mod 7.

Note that the Corollary applies to 1/2 of all pairs (k, p). Both the Corollary and Theorem 1 follow from the factorizations in Theorem 2 and the following proposition.

Proposition 2. Suppose T_{n,k}(x) is irreducible for some n. Then for each m we have T_{m,k}(x) = f(x)^{r} with f(x) irreducible and r ∈ ℕ. Suppose T_{n,k}(x) is irreducible and has full Galois group for some n. Then for each m either T_{m,k}(x) is irreducible and has full Galois group, or T_{m,k}(x) = (x − a)^{d_{k}} for some a ∈ ℤ.

Proof of the Corollary and Theorem 1. If T_{p,k}(x) = f(x)^{r} then each root of T_{p,k}(x) mod ℓ has multiplicity divisible by r. Using ℓ = 5 or 7 and the factorizations in Theorem 2a gives r = 1 for all cases covered by the Corollary.

If T_{p,k}(x) = (x − a)^{r} then T_{p,k}(x) has only one root mod ℓ. By Theorem 2a this does not hold in the cases covered by Theorem 1.

Sketch of Proof of Proposition 2. The idea is to consider how the Galois group of the field of Fourier coefficients of the cusp forms acts on the Hecke basis.

Let B_{k} be the set of normalized Hecke eigenforms in S_{k}(1), let 𝕂_{k}/ℚ be the field generated by the set of Fourier coefficients of the f ∈ B_{k}, and let G = Gal(𝕂_{k}/ℚ). If f(z) = ∑ a_{n}q^{n} with {a_{n}} ⊂ 𝕂_{k} and σ ∈ G then set σf(z) = ∑ σa_{n}q^{n}.

Since S_{k}(1) has a rational basis, G acts on S_{k}(1). The group G also acts on the set B_{k} because G commutes with the Hecke operators. This induces an action on the roots of T_{n,k}(x) because those roots are the nth Fourier of the Hecke eigenforms. That action is the same as the natural action given by the inclusion of those roots in 𝕂_{k}. Thus, if G acts transitively on B_{k} then G acts transitively on the roots of T_{m,k}(x), which implies the first conclusion in the Proposition. If G acts as the full symmetric group on B_{k}, then G acts as the full symmetric group on the roots of T_{m,k}(x), which implies the second conclusion in the Proposition.

To finish the proof, note that the Galois group of T_{n,k}(x) acts on B_{k} by extending the action on the nth Fourier coefficients. Thus, if T_{n,k}(x) is irreducible then 𝕂_{k} is the splitting field of T_{n,k}(x) and G is its Galois group. This proves Proposition 2.

It would be interesting to prove a version of Maeda’s conjecture of the form “if T_{n,k}(x) is irreducible for some n, then it is irreducible for all n,” or even “if T_{n,k}(x) is irreducible for some n, then T_{2,k}(x) is irreducible.” The methods used here are only able to establish such a result for a large proportion of weights k. By combining Theorem 2b with the Corollary above, we have that if T_{n,k}(x) is irreducible for some n and dim S_{k}(1) is not a multiple of 14, then T_{2,k}(x) is irreducible, and if dim S_{k}(1) is not a multiple of 28, then T_{2,k}(x) or T_{3,k}(x) is irreducible. Establishing factorizations of T_{2,k}(x) mod ℓ for larger ℓ would increase the proportion of k for which such a result is known. This is reminiscent of Lehmer’s conjecture that the Ramanujan τ–function never vanishes, where the results in that direction come from congruence properties of τ(n).

References

[B] K. Buzzard, On the eigenvalues of the Hecke operator T_{2}, J. Number Theory 57 (1996), no. 1, 130–132.

[C] H. Cohen, Trace des opérateurs de Hecke sur Γ_{0}(N), in “Séminaire de Théorie des nombres,” exp. no. 4, Bordeaux, 1976-1977.

[CF] J.B. Conrey and D.W. Farmer, Hecke operators and the nonvanishing of L–functions, to appear in the proceedings of the Penn State conference, Summer 1997.

[FJ] D.W. Farmer and K. James, The irreducibility of some level 1 Hecke polynomials, preprint.

[HM] H. Hida and Y. Maeda, Non–abelian base change for totally real fields, Pac. J. Math 181 no. 3 (1997) 189–218.

[JO] K. James and K. Ono, On the factorization of Hecke polynomials, preprint.

[J] N. Jochnowitz, Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math.

Soc. 270 (1982), 269–285.

[KZ] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175–198.

[Ser] J–P Serre, Valeurs propres des opérateurs de Hecke modulo ℓ. Astérisque 24–25 (1975), 109–117.

[SV] R. Schoof and M. van der Vlugt, Hecke operators and the weight distribution of certain codes, Journal of Combinatorial Theory A, 75 (1991), 163–186.