HECKE OPERATORS AND THE

NONVANISHING OF L–FUNCTIONS

J.B. Conrey D.W. Farmer

Abstract. We determine some conditions which imply the nonvanishing of L_{f} (k/2) for f a cusp form of weight k ≡ 0 mod 4 for the full modular group. These conditions are verified for weights k ≤ 500.

1. Introduction and statement of results

Let S_{k}(1) be the space of holomorphic cusp forms of even integral weight k for the full modular group. This space has a basis of Hecke eigenforms

B = \{f_{1}, …, f_{d}\}, d = \{ \begin{matrix}[\frac{k}{12}] − 1 & if k ≡ 2 mod 12; \\ [\frac{k}{12}] & otherwise, \end{matrix}

satisfying T_{n}f_{j} = a_{j}(n)f_{j} for all n, where T_{n} is the n^{th} Hecke operator. The f_{j} have Fourier expansions

f_{j}(z) = ∑_{n=1}^{∞} a_{j}(n)e(nz)

where e(z) = exp(2πiz). We normalize f_{j} by setting a_{j}(1) = 1.

A Dirichlet series is associated to f_{j} by

L_{f_{j}}(s) = ∑_{n=1}^{∞} \frac{a_{j}(n)}{n^{s}}.

In this paper we study the “critical value” L_{f_{j}}(k/2). Since L_{f_{j}}(s) has functional equation

(2π)^{−s}Γ(s)L_{f_{j}}(s) = (−1)^{k/2}(2π)^{s−k}Γ(k − s)L_{f_{j}}(k − s),

it follows that L_{f_{j}}(k/2) = 0 if k ≡ 2 mod 4. If k ≡ 0 mod 4, then L_{f_{j}}(k/2) = 0 implies that L_{f_{j}}(s) has a zero of order at least 2 at s = k/2. In this paper we make progress toward showing that L_{f_{j}}(k/2) ≠ 0 for k ≡ 0 mod 4.

Typeset by AMS-TEX

Research of the first author supported in part by a grant from the NSF.

Our motivation for this work came from a recent conjecture of Duke and Imamoglu [DI] concerning Siegel modular forms. If n and k are even integers and f is a classical cusp form of weight 2k − n, they conjecture that there is a Siegel cusp form F of degree n and weight k whose standard L–function factors as

L(F, s) = ξ(s) ∏_{j=1}^{n−1} L(f, s − j/2)L(f, s + j/2),

where all L–functions have been normalized to have functional equations sending s ↦ 1 − s. If L_{f}(s) vanishes at the critical point then the above expression does not have a pole at s = 1. Thus, the vanishing of L_{f}(s) is related to whether or not the Siegel modular form F is a linear combination of theta series [Bo].

Our main result is

Theorem 1. Suppose k ≡ 0 mod 4. If, for some n, the characteristic polynomial of T_{n} acting on S_{k}(1) is irreducible over ℚ, then L_{f_{j}}(k/2) ≠ 0 for all j.

Maeda has conjectured that the Hecke algebra of SL_{2}(ℤ) is simple, which implies that the condition in Theorem 1 always holds. It appears to be possible that the characteristic polynomial is irreducible for all n. Theorem 1 is also stated by Kohnen and Zagier [KZ].

We have verified the irreducibility of the characteristic polynomial of T_{2} acting on S_{k}(1) for k ≤ 500, yielding

Theorem 2. L_{f_{j}}(k/2) ≠ 0 for k ≤ 500, k ≡ 0 mod 4.

Questions concerning the characteristic polynomials of Hecke operators have received considerable attention lately. For example, if N is large then the characteristic polynomial of T_{p} acting on S_{2}(Γ_{0}(N)) will have lots of linear factors corresponding to elliptic curves, but one can look for irreducible factors of a large degree (see [K]). At the other extreme, one expects the characteristic polynomial of T_{p} acting on S_{k}(Γ(1)) to be irreducible, so Buzzard [B] began a study of its Galois group. We applied the method of Buzzard to show

Theorem 3. For k ≤ 500, k ≡ 0 mod 4, let T_{2,k}(x) ∈ ℤ[x] be the characteristic polynomial of T_{2} acting on S_{k}(1). Then the Galois group of the splitting field of T_{2,k}(x) over ℚ is the symmetric group S_{d}.

Corollary 4. For k ≤ 500, k ≡ 0 mod 4, let 𝕂_{k} be the field obtained by adjoining to ℚ all of the Fourier coefficients of all of the Hecke eigenforms f_{j}. Then 𝕂_{k}/ℚ is a Galois extension of degree d with Gal(𝕂_{k}/ℚ) = S_{d}.

2. Background and Notation

Our methods involve the Hecke operators T_{n}. These can be defined as linear maps T_{n} : S_{k}(1) → S_{k}(1) given by

T_{n} ( ∑_{m=1}^{∞} a(m)e(mz)) = ∑_{m=1}^{∞} ( ∑_{d|(n,m)} d^{k−1}a (\frac{mn}{d^{2}})) e(mz)

There is an inner product on S_{k}(1), called the Petersson inner product,

⟨f, g⟩ = ∫∫_{D} f(z)\overline{g(z)}y^{k}dμ(z),

where D is a fundamental domain for the full modular group, and dμ(z) = y^{−2}dxdy. It can be shown that the Hecke operators mutually commute and are self–adjoint with respect to the Petersson inner product. Thus there is a basis of simultaneous eigenfunctions of the T_{n}, which we have already denoted by B = {f_{1}, …, f_{d}}.

The holomorphic Eisenstein series E_{k}(z) ∈ M_{k}(1) is defined by

E_{k}(z) = \frac{1}{2ζ(k)} ∑_{γ∈Γ_{∞}\Γ} j(γ, z)^{−k} = 1 − \frac{2k}{B_{k}} ∑_{m=1}^{∞} σ_{k−1}(m)e(mz),

where j(γ, z) = cz + d if γ is a 2 × 2 matrix with bottom row (c d), the B_{k} are Bernoulli numbers, and

σ_{ℓ}(n) = ∑_{d|n} d^{ℓ}.

The Eisenstein series E_{k}(z) is also an eigenfunction of the T_{n}, and B ∪ {E_{k}} is a basis for M_{k}(1). For more information on modular forms for the full modular group, see Apostol’s book [A].

We make extensive use of the fact that S_{k}(1) has a basis of functions with rational integer Fourier coefficients. Set B_{ℚ} = {F_{1}, …, F_{d}} where

F_{j} = Δ^{j}E_{4}^{\frac{k}{4} −3j} = ∑_{m=j}^{∞} A_{j}(m)e(mz),  A_{j}(m) ∈ ℤ,  (2.1)

say, where Δ ∈ S_{12}(1) is the cusp form of weight 12. We denote by T_{n,k}(x) the characteristic polynomial of T_{n} acting on S_{k}(1). The matrix representing T_{n} in the basis B_{ℚ} has rational entries, so T_{n,k}(x) ∈ ℤ[x].

We require some information about the field 𝕂_{k} obtained by adjoining to ℚ all of the Fourier coefficients a_{j}(n) of all of the f_{j}. This is the same as the splitting field of all of the T_{n,k}(x).

Lemma 1. 𝕂_{k} = ℚ({a_{j}(n) : 1 ≤ j, n ≤ d}). In particular, 𝕂_{k} is a finite Galois extension of ℚ.

Proof. Since B_{ℚ} is a basis, we can write

f_{j} = ∑_{1≤m≤d} α_{j,m}F_{m}  (2.2)

The α_{j,m} can be found by solving the “upper triangular” system

a_{j}(n) = ∑_{1≤m≤d} α_{j,m}A_{m}(n), 1 ≤ j, n ≤ d.

Thus, the α_{j,m} are in the field ℚ({a_{j}(n) : 1 ≤ j, n ≤ d}). But then by (2.2) all of the Fourier coefficients of the f_{j} are in that field, so 𝕂_{k} is as described.

The Galois group Gal(𝕂_{k}/ℚ) acts on functions with Fourier coefficients in 𝕂_{k} by

σ ( ∑_{m=1}^{∞} a(m)e(mz)) = ( ∑_{m=1}^{∞} σ (a(m))e(mz)) .

Lemma 2. Gal(𝕂_{k}/ℚ) acts on B = {f_{1}, …, f_{d}}. If T_{n,k}(x) is irreducible for some n, then the action is transitive. Furthermore, if T_{n,k}(x) is irreducible then 𝕂_{k}/ℚ is the splitting field of T_{n,k}(x) and Gal(𝕂_{k}/ℚ) = Gal(T_{n,k}).

Proof. Gal(𝕂_{k}/ℚ) acts on B because the elements of Gal(𝕂_{k}/ℚ) commute with each Hecke operator T_{n}, and the f_{j} are the simultaneous eigenfunctions of the T_{n}.

If T_{n,k}(x) is irreducible then Gal(𝕂_{k}/ℚ) acts transitively on {a_{j}(n) : 1 ≤ j ≤ d}, which is the set of n^{th} Fourier coefficients of the f_{j}. This forces the action to be transitive on B. Furthermore, Gal(𝕂_{k}/ℚ) ⊂ Gal(T_{n,k}) by considering the action of Gal(𝕂_{k}/ℚ) on B and restricting to the set of n^{th} Fourier coefficients. By definition the splitting field of T_{n,k}(x) is contained in 𝕂_{k}, so they must be equal.

In the next section we relate the transitive action of Gal(𝕂_{k}/ℚ) on B to the nonvanishing of L_{f_{j}}(k/2). In the following section we describe our calculations which show that T_{2,k}(x) is irreducible for small k.

3. Proof of Theorem 1

Let

\begin{aligned}f(z) & = ∑_{m=1}^{∞} a(m)e(mz) \\ & = f_{j}(z), \end{aligned}

for some j. Set

F (s, ℓ) = ∑_{n=1}^{∞} \frac{a(n)σ_{ℓ−1}(n)}{n^{s}}.

The next two propositions make use of ideas of Shimura [Sh], and can also be found in [Z].

Proposition 1. We have F (k − 1, k/2) = 0 if and only if L_{f}(k/2) = 0.

Proof. By comparing Euler products, we find that

F (s, ℓ) = \frac{L_{f}(s)L_{f}(s + 1 − ℓ)}{ζ(2s + 2 − k − ℓ)}.

(For a discussion of this method, see Bump’s article [Bu].) In particular,

F (k − 1, k/2) = \frac{L_{f}(k − 1)L_{f}(k/2)}{ζ(k/2)}.

To finish the proof, note that k − 1 is strictly within the plane of absolute convergence for the Euler product for L_{f}(s), so L_{f}(k − 1) ≠ 0.

Proposition 2. We have ⟨E_{k/2}^{2}, f⟩ = C_{k} L_{f}(k − 1)L_{f}(k/2), where

C_{k} = \frac{(−1)^{\frac{k}{4}}}{\sqrt{2}} (\frac{2}{π} )^{\frac{k−1}{2}} \frac{Γ (\frac{k−1}{2})}{ζ(\frac{k}{2} )^{3}}.

In particular, ⟨E_{k/2}^{2}, f⟩ = 0 if and only if L_{f}(k/2) = 0.

Proof. Let

I(s) = ∫_{0}^{∞} ∫_{0}^{1} \overline{f(z)}E_{ℓ}(z)y^{s−1} dx dy.

By substituting the Fourier series for E_{ℓ}(z) and f(z) into the equation for I(s) and integrating term–by–term, we find that

I(s) = −\frac{2ℓ}{B_{ℓ}} (4π)^{−s}Γ(s)F (s, ℓ).

But by the Rankin–Selberg unfolding method [Bu], we also have

I(s) = ∫∫_{D} \overline{f(z)}E_{ℓ}(z) ∑_{γ∈Γ_{∞}\Γ} \overline{j(γ, z)}^{k}j(γ, z)^{ℓ}y(γz)^{s+1}dμ(z),

where D is a fundamental domain for the full modular group. The sum over γ equals 2ζ(k−ℓ)y(z)^{k}E_{k−ℓ}(s+1−k, z), which involves the non-holomorphic Eisenstein series defined by

E_{k}(s, z) = \frac{1}{2ζ(k)} ∑_{γ∈Γ∞\Γ} j(γ, z)^{−k}y(γz)^{s},

where y(u + iv) = v. In the notation of the Petersson inner product, we have

−\frac{2ℓ}{B_{ℓ}} (4π)^{−s}Γ(s)F (s, ℓ) = 2ζ(k − ℓ)⟨E_{ℓ}(·)E_{k−ℓ}(s + 1 − k, ·), f(·)⟩.

If we take s = k − 1 and ℓ = k/2, and use the fact that E_{k/2}(0, z) = E_{k/2}(z), we obtain the equation in the proposition, after expressing B_{k/2} in terms of ζ(k/2) and using the duplication formula for the Γ–function.

Proof of Theorem 1. A basis for the space of modular forms of weight k is

{f_{1}, f_{2}, …, f_{d}, E_{k}}.

Since E_{k/2}^{2} is a modular form of weight k, there is a unique way to write

E_{k/2}^{2} = E_{k} + ∑_{1≤j≤d} c_{j}f_{j}  (3.1)

for some coefficients c_{j} ∈ 𝕂_{k}. In particular,

⟨E_{k/2}^{2}, f_{j}⟩ = c_{j}⟨f_{j}, f_{j}⟩,

so by Proposition 2, L_{f_{j}}(k/2) = 0 if and only if c_{j} = 0.

Applying σ ∈ Gal(𝕂_{k}/ℚ) to (3.1) gives

E_{k/2}^{2} = E_{k} + ∑_{1≤j≤d} σ(c_{j})f_{\overline{σ}(j)},

where \overline{σ} is the permutation of \{1, 2, …, d\} induced by the action of σ on B = \{f_{1}, …, f_{d}\}. Since the expression (3.1) is unique, it follows that σ(c_{j}) = c_{\overline{σ}(j)}.

Suppose Gal(𝕂_{k}/ℚ) acts transitively on B, and c_{j}=0 for some j. Then by the previous observation we find that c_{j} = 0 for all j. This implies E_{k/2}^{2} = E_{k}. This is a contradiction if k > 8, as can be seen by looking at the second Fourier coefficient of E_{k} and E_{k/2}^{2} and using elementary properties of the Bernoulli numbers. So by Lemma 2, if T_{n,k}(x) is irreducible for some n, then c_{j} ≠ 0. This proves Theorem 1.

Note: In the above proof we required the fact that E_{k/2}^{2} = E_{k} does not hold for k > 8. E. Ghate [G] has used the ideas in this paper to give an interesting proof that the only identities among Eisenstein series are the classical ones: E_{4}^{2} = E_{8}, E_{4}E_{6} = E_{10}, and E_{6}E_{8} = E_{14}.

4. Calculation of the Hecke operators

In this section we describe some calculations involving the Hecke operator T_{2} acting on S_{k}(1). Our goal is to show that the the characteristic polynomial T_{2,k}(x) of T_{2} is irreducible and has Galois group S_{d}, where d = [k/12] = dim(S_{k}(1)).

Direct calculation of the polynomial T_{2,k}(x) is difficult for large k because of the huge size of the coefficients (T_{2,k}(x) is a polynomial of degree d, with roots of size 2^{6d}). To verify that T_{2,k}(x) is irreducible we calculate T_{2,k}(x) mod p for small p and check if the reduced polynomial is irreducible mod p. Having established that the polynomial is irreducible, we use the following extension of the method of Buzzard [B] to show that it has Galois group S_{d}.

Lemma 3. Suppose G is a transitive subgroup of the symmetric group S_{d}. If G contains both a transposition and a p–cycle for some prime p > d/2, then G = S_{d}.

Proof. Put a relation ∼ on S = {1, 2, …, d} by a ∼ b if a = b or if the transposition (a b) ∈ G. This is easily seen to be an equivalence relation. Since G is transitive, each equivalence class has the same number of elements, say n, and it follows that n|d. Note that n > 1 because G contains at least one transposition.

Let T = {e_{1}, e_{2}, …, e_{p}} be the subset of S permuted by the assumed p–cycle in G, and let G_{T} be the subgroup of G fixing the elements not in T. Define an equivalence relation on T by a ≈ b if a = b or if the transposition (a b) ∈ G_{T}. As above, each equivalence class has the same number of elements, say m, and m|p. Since n > 1, we have m > 1, so m = p because p is prime. But n ≥ m because G_{T} ⊂ G. Thus n > d/2, so n = d. This implies G = S_{d}.

Lemma 4. Let f(x) ∈ ℤ[x] be a monic irreducible polynomial of degree d, with splitting field 𝕂/ℚ. Suppose there is a prime q such that the mod q reduction f_{q} ∈ 𝔽_{q}[x] satisfies

f_{q} = g_{0}g_{1} ⋯ g_{j}

for distinct irreducible g_{i} ∈ 𝔽_{q}[x] with

\begin{matrix}deg(g_{0}) = 2 \\ deg(g_{i}) odd for i ≥ 1. \end{matrix}

Suppose there also is a prime r with the mod r reduction f_{r} ∈ 𝔽_{r}[x] satisfying

f_{r} = h_{0}h_{1} ⋯ h_{l}

for distinct irreducible h_{i} ∈ 𝔽_{r}[x] with deg(h_{0}) = p, with p > d/2 prime. Then Gal(𝕂/ℚ) is the full symmetric group S_{d}.

Proof. Identify G = Gal(𝕂/ℚ) as a subgroup of S_{d}. We have assumed that f(x) has distinct roots mod q and mod r. So if Q and R are primes of 𝕂 above q and r, respectively, then we have Frob_{Q}, Frob_{R} ∈ G. Appropriate powers Frob_{Q}^{t_{1}} and Frob_{R}^{t_{2}} are a transposition and a p–cycle, respectively. Lemma 3 applies, so G = S_{d}.

The following is our procedure to check that T_{2,k}(x) is irreducible and has Galois group S_{d}.

Step 1. Generate the basis B_{ℚ} = {F_{1}, …, F_{d}} given in (2.1). We use the first 200 terms in the Fourier expansion.

Step 2. Form the matrices A = (a_{j,l}) with a_{j,l} = A_{j}(2l) + 2^{k−1}A_{j}(l/2) and B = (A_{j}(l)), where A_{j}(l) is given in (2.1).

Note that AB^{−1} is the matrix of T_{2} in the basis B_{ℚ}, and det(A − xB) ∈ ℤ[x] is the characteristic polynomial of T_{2}.

Step 3. Reduce A − xB mod p, compute the determinant of the reduced matrix, and factor that polynomial mod p.

Step 3 is repeated until we find a prime p for which the polynomial is irreducible, and primes q and r as in Lemma 4.

The above procedure was implemented in Mathematica on a Sun SparcCenter 1000 with 4 processors. The program took 867 CPU minutes for the range 24 ≤ k ≤ 400, and an additional 2103 minutes for the range 404 ≤ k ≤ 500. It was found that for all k in those ranges the characteristic polynomial T_{2,k}(x) is irreducible and has Galois group S_{d}.

Combining the computer calculation with Theorem 1 proves Theorem 2. Combining the computer calculation with Lemma 4 proves Theorem 3. Corollary 4 follows because, by Lemma 2, if T_{2,k}(x) is irreducible then 𝕂_{k} = ℚ({a_{j}(2) : 1 ≤ j ≤ d}).

The results of the calculation, for small k, are given in the table below. The table shows the smallest primes p, q, r for which the polynomial T_{2,k}(x) is irreducible mod p, and q, r are as in Lemma 4. (If d is prime, then we put r = p).

\begin{matrix}k & p & q & r \\ 24 & 23 & 23 & 23 \\ 28 & 29 & 29 & 29 \\ 32 & 29 & 29 & 29 \\ 36 & 37 & 29 & 37 \\ 40 & 37 & 41 & 37 \\ 44 & 61 & 41 & 61 \\ 48 & 53 & 23 & 43 \\ 52 & 67 & 29 & 53 \\ 56 & 71 & 29 & 47 \\ 60 & 83 & 37 & 83 \\ 64 & 73 & 29 & 73 \\ 68 & 97 & 23 & 97 \\ 72 & 107 & 23 & 73 \\ 76 & 137 & 47 & 73 \\ 80 & 103 & 29 & 89 \end{matrix}  \begin{matrix}k & p & q & r \\ 84 & 307 & 37 & 307 \\ 88 & 349 & 31 & 349 \\ 92 & 107 & 41 & 107 \\ 96 & 149 & 23 & 101 \\ 100 & 179 & 41 & 89 \\ 104 & 127 & 37 & 89 \\ 108 & 223 & 29 & 83 \\ 112 & 127 & 23 & 107 \\ 116 & 263 & 53 & 107 \\ 120 & 293 & 23 & 113 \\ 124 & 163 & 29 & 107 \\ 128 & 149 & 59 & 127 \\ 132 & 479 & 37 & 479 \\ 136 & 139 & 29 & 139 \\ 140 & 149 & 41 & 149 \end{matrix}

References

[A] T. Apostol, Modular functions and Dirichlet series in number theory, Springer–Verlag GTM

41, 1990.

[Bo] S. Böcherer, Siegel modular forms and theta series, Theta Functions, Bowdoin 1987, AMS

Proceedings of Symposia in Pure Mthematics, vol. 49, 3–17.

[Bu] D. Bump, The Rankin-Selberg method: a survey, in Number theory, trace formulas and discrete

groups (Oslo, 1987), 49–109, Academic Press, Boston, MA, 1989.

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

1, 130–132.

[DIm] W. Duke and O. Imamoglu, personal communication.

[G] E. Ghate, On monomial relations between Eisenstein series, preprint.

[K] S. Kamienny, On the eigenvalues of Hecke operators, Comm. Algebra 23 (1995), no. 3, 995–998.

[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.

[Sh] G. Shimura, The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math. 9 (1976), no. 6, 783–804.

[Z] D. Zagier, Modular forms whose Fourier coefficients involve zeta–functions of quadratic fields, Modular Functions of One Variable VI, LNM 627, 105–170.