IRREDUCIBILITY OF HECKE POLYNOMIALS

SRINATH BABA AND M. RAM MURTY

Abstract. In this note, we show that if the characteristic polynomial of some Hecke operator T_{n} acting on the space of weight k cusp forms for the group SL_{2}(ℤ) is irreducible, then the same holds for T_{p}, where p runs through a density one set of primes. This proves that if Maeda’s conjecture is true for some T_{n}, then it is true for T_{p} for almost all primes p.

1. Introduction

Let V be the d-dimensional space of weight k cusp forms S_{k}(SL_{2}(ℤ)), and T_{n} the n^{th} Hecke operator on V. Let T_{n}(x) denote the characteristic polynomial of T_{n}. By the theory of eigenforms, it is well known that T_{n}(x) ∈ ℤ[x] and is monic. Maeda [8] conjectured that for some n, T_{n}(x) is irreducible with Galois group S_{d}, where S_{d} is the symmetric group on d symbols. A popular extension of this conjecture, called Maeda’s conjecture states that forevery n, T_{n}(x) is irreducible with Galois group S_{d}.

Recent progress related to Maeda’s conjecture has been in two different directions– the first is to verify the conjecture for T_{2}(x) for different weights k ([1], [12]), and the second has been to show irreducibility of T_{p}(x) assuming the irreducibility of T_{q}(x) for some q. In [2], it is shown using the trace formula in characteristic p that if some T_{q}(x) satisfies Maeda’s conjecture, then the same holds for T_{p}(x) for p in a set of primes of density 5/6. Combining this with computer computations, K. James and D. Farmer have shown in [3] that if T_{q}(x) satisfies Maeda’s conjecture for some q, then the same holds for T_{p}(x) for primes p ≤ 2000.

The purpose of this note is to extend the work of both [2] and [3]. As opposed to the mod-p versions of the trace formula used by [2], we study Frobenius distributions and Galois representations of Hecke eigenforms to show that

Theorem 1.1. If T_{q}(x) is irreducible for some prime q, then

♯{p ≤ x; T_{p}(x) is reducible} ≪ x/(log x)^{1+δ}

for some δ > 0 .

In addition, we show the following:

Theorem 1.2. If T_{q}(x) is irreducible with Galois group S_{d} for some prime q, then the same holds for T_{n}(x) for n ≤ d.

The argument is based on the existence of a so called Miller basis for V, and uses only linear algebra.

2. Preliminaries

For any cusp form g , let g = ∑_{i=1}^{∞} a_{n}(g)q^{n} denote its Fourier expansion at the cusp at ∞. Let K(g) = ℚ({a_{i}(g)}_{i=1}^{∞}). Let f_{1}, … f_{d} denote a basis of normalised Hecke eigenforms for V. For any i, 1 ≤ i ≤ d, it is known that K(f_{i}) is a number field of finite degree, and that the a_{n} are all algebraic integers. Since the f_{i} are simultaneous eigenvectors of all the Hecke operators, we know that T_{n}(x) = ∏_{i=1}^{d}(x−a_{n}(f_{i})).

Lemma 2.1. Suppose T_{q}(x) is irreducible, where q is a prime. Then K(f_{i}) = ℚ(a_{q}(f_{i})).

Proof. Let h_{1}, … h_{d} be the Miller basis for V. This basis is characterised by the property that K(h_{i}) = ℚ for every i, and a_{i}(h_{j}) = δ_{i,j}, for 1 ≤ i, j ≤ d, where δ_{i,j} = 1 if i = j and 0 otherwise (see [6]). Let f be any of the eigenforms. Expressing f as a linear combination of the h_{i}, we have

f = ∑_{i=1}^{d} a_{i}(f)h_{i} = h_{1} + ∑_{i=2}^{d} a_{i}(f)h_{i}.

Since T_{q}(x) is irreducible, it is immediate that not all of the coefficients a_{q}(h_{2}), …, a_{q}(h_{d}) are 0. Without loss of generality, we assume that a_{q}(h_{2}) ≠ 0. By row reducing the Miller basis, we can obtain a basis g_{1} … g_{d} so that a_{1}(g_{1}) = 1, a_{1}(g_{j}) = 0 for j ≠ 1, and a_{q}(g_{2}) = 1, a_{q}(g_{j}) = 0 for j ≠ 2, and K(g_{j}) = ℚ for every g_{j}.

Expressing f in terms of the new basis, we realise that one of the Fourier coefficients a_{q^{2}}(g_{3}), …, a_{q^{2}}(g_{d}) must be non-zero. By repeating the row reduction argument and producing a new basis each time, we construct a basis F_{1}, … F_{d} with the property that a_{q^{i−1}}(F_{j}) = δ_{i,j}. In addition, K(F_{i}) = ℚ for every F_{i}. Expressing f in terms of this basis, and observing that a_{q^{l}}(f) ∈ ℚ(a_{q}(f)) for every l ∈ ℤ, we have

f = ∑_{i=1}^{d} a_{q^{i−1}}(f)F_{i} = ∑_{i=1}^{d} b_{i}F_{i} where b_{i} ∈ ℚ(a_{q})

Thus, for any n, a_{n}(f) ∈ ℚ(a_{q}(f)), and so the lemma follows.

Lemma 2.2. If T_{q}(x) is irreducible for some prime q with Galois group G, then for any other n, T_{n}(x) has exactly one irreducible factor. In addition, if G = S_{d}, then the irreducible factor of T_{n} has degree d or 1.

Proof. Suppose T_{q}(x) is irreducible with Galois group G. Then G acts transitively on the roots of T_{q}(x). Since the roots of T_{n}(x) by Lemma [2.1], are rational linear combinations of the roots of T_{q}(x), they form a Galois orbit with a G action. Thus, they are all roots of the same irreducible polynomial. If, in addition, G = S_{d}, then the roots of T_{n}(x) must form an orbit for a transitive S_{d} action, and thus must all be equal, or all distinct. This proves the lemma.

Consider a Galois extension L/ℚ with Galois group G discriminant d_{L} and degree n_{L}. Suppose S is the set of primes in K ramified in L. Let C be a collection of conjugacy classes in G. Let

π_{C}(x) = the number of primes v ∈ O_{L}; N_{L/ℚ}(v) ≤ x and Frob_{v} ∈ C

The Chebotarev density theorem states that

π_{C}(x) ∼ \frac{|C|}{|G|}π(x)

where π(x) = the number of primes p ∈ ℤ; p ≤ x.

The following unconditional effective version of this theorem was provided by Lagarias, Montgomery and Odlyzko in [7], we state a version due to Serre( see [11], Theorem 3, page 132).

Proposition 2.3 (Effective version of the Chebotarev density theorem). If x ≥ 3 and log x ≥ c(log d_{L})(log log d_{L})(log log log 6d_{L})), then π_{C}(x) ≪ \frac{|C|}{|G|}π(x) where c is an absolute constant.

We can bound the discriminant d_{L} of L by the following (see [11] Page 129, Proposition 4’ for a general statement)

Proposition 2.4 (Hensel). log d_{L} ≤ (n_{L} − 1) ∑_{l∈S} log l + n_{L} log n_{L}

3. Fourier coefficients of Hecke eigenforms

Let f be a normalised Hecke eigenform, and suppose that for some q, [ℚ(a_{q}(f)) : ℚ] = d. This is the same as saying T_{q}(x) is irreducible. From Lemma2.1, we know that K(f) = ℚ(a_{q}(f)).

Theorem 3.1. Let L ⊂ K(f) be any proper subfield. Then,

♯\{p ≤ x; a_{p}(f) ∈ L\} ≪ \frac{x}{(log x)^{1+δ}}

for some δ > 0.

Proof. Let K(f) = K, and let L ⊂ K(f) be a proper subfield. Let λ ∈ O_{K} be a prime of degree f ≥ 2 lying above l ∈ O_{L}. By a well-known construction of Deligne and Serre (see [10], Pages 260-261), there exists a continuous representation

ρ_{f,λ} : Gal(ℚ̄/ℚ) ⟶ GL_{2}(O_{K}/λ)

satisfying the following conditions for p ≠ l:

(i) ρ_{f,λ} is unramified at p (ii) trace(ρ_{f,λ}(Frob_{π})) = a_{p}(f), for any prime ideal π lying above p.

Let S ⊂ GL_{2}(O_{K}/λ) be the set of elements whose trace lies in the subfield O_{L}/l ⊂ O_{K}/λ. By a simple counting argument, we see that

♯(S) ≪ l^{3f+1} and ♯(GL_{2}(O_{K}/λ)) ∼ l^{4f}.

Let M be the fixed field of the kernel of the representation ρ_{f,λ}. By the effective version of the Chebotarev density theorem in M, if l ∼ (log x)^{δ}, then

♯\{p ≤ x; trace(ρ_{f,λ}(Frob_{p})) ∈ O_{L}/l ⊂ O_{K}/λ\} ≪ \frac{♯(S)}{♯(GL_{2}(O_{K}/λ))}𝔵/log x

By the bounds on ♯(S) and ♯(GL_{2}(O_{K}/λ)) and on l, we see that

♯\{p ≤ x; trace(ρ_{f,λ}(Frob_{p})) ∈ O_{L}/l ⊂ O_{K}/λ\} ≪ \frac{x}{l^{f−1} log x} ≪ \frac{x}{(log x)^{1+δ}}

This proves the theorem.

Proof of Theorem 1.1. By Lemma 2.2, if T_{q}(x) is irreducible, then T_{p}(x) is reducible if and only if it has a repeated root. Thus, K(f_{1}) contains ℚ(a_{p}(f_{1})) as a proper subfield. Since there are only finitely many proper subields of K(f_{1}), we can apply Theorem 3.1 to each subfield. Thus, we see that

♯\{p ≤ x; T_{p}(x) is reducible \} ≪ \frac{x}{(log x)^{1+δ}}

This proves the theorem.

4. Initial Fourier coefficients

Proof of Theorem 1.2. By Lemma 2.2, we know that T_{n}(x) is reducible only if it has a single repeated root, i.e., a_{n}(f_{1}) = ⋯ a_{n}(f_{d}) = a ∈ ℤ. Suppose this holds for some i, 2 ≤ i ≤ d. Let h_{i} be as in Lemma 2.1. Let h_{i} be written as a linear combination of the eigenforms as

h_{i} = ∑_{i=1}^{d} c_{i,j}f_{j}

Since a_{1}(h_{i}) = 0 and a_{1}(f_{j}) = 1 for every f_{j}, we conclude that ∑_{i=i}^{d} c_{i,j} = 0. Thus,

1 = a_{i}(h_{i}) = ∑_{i=1}^{d} c_{i,j}a_{i}(f_{j}) = ∑_{i=1}^{d} c_{i,j}a = 0

which shows us that our assumption is false. This proves the Theorem.

5. Comparison Theorems for Fourier coefficients of two eigenforms

Let f and g be two distinct Hecke eigenforms for SL_{2}(ℤ) of weights k_{1} and k_{2} respectively. If k_{1} = k_{2}, then Theorem 1.1 implies the following:

Theorem 5.1. If T_{q}(x) is irreducible for some prime q, then

♯\{p ≤ x; a_{p}(f) = a_{p}(g)\} ≪ \frac{x}{(log x)^{1+δ}}

for some δ > 0

In the case k_{1} ≠ k_{2}, we can prove a similar result unconditionally.

Theorem 5.2. If k_{1} ≠ k_{2}, then

♯\{p ≤ x; a_{p}(f) = a_{p}(g)\} ≪ \frac{x}{(log x)^{1+δ}}

for some δ > 0

In [10], section 5, Ribet studied pairs of Galois representations and showed that if l is sufficiently large, the two Galois representations ρ_{f,λ} and ρ_{g,λ} are “as independent as possible”, i.e., the image of the product representation is as large as possible. In the case f and g are as in Theorem 5.2, we state Ribet’s theorem as follows:

Lemma 5.3. Im(ρ_{f,λ}, ρ_{g,λ}) = {(u, u′) ∈ GL_{2}(O_{λ}) × GL_{2}(O_{λ}); det(u) = v^{k_{1}−1}; det(u′) = v^{k_{2}−1}, v ∈ O_{λ}}

Let F_{λ} = O_{λ}/λ. If we let ρ̄_{f,λ} : Gal(ℚ̄/ℚ) ⟶ GL_{2}(F_{λ}) denote the residual representation of ρ_{f,λ}, we see that

Im(ρ̄_{f,λ}, ρ̄_{g,λ}) =
{(u, u′) ∈ GL_{2}(F_{λ}) × GL_{2}(F_{λ}); det(u) = v^{k_{1}−1}; det(u′) = v^{k_{2}−1}, v ∈ F_{λ}}

Proof of Theorem 5.2. Let E/ℚ be an extension containing both E_{f} and E_{g}, and λ a degree one prime in E of norm l (by the prime number theorem in number fields, we know that degree 1 primes are of full density, see [6], Theorem 4, Page 350). Let

S = {(u, u′) ∈ GL_{2}(F_{λ}) × GL_{2}(F_{λ}); det(u) = v^{k_{1}−1}, det(u′) = v^{k_{2}−1}, v ∈ F_{λ}}

S′ = {(u, u′) ∈ S; trace(u) = trace(u′)}

|S| ≤ l^{7} and |S′|/|S| ≤ 1/l. (1)

By the above, we know that the image of the product representation ρ̄_{f,λ} × ρ̄_{g,λ} is exactly the group S. Let K be the fixed field of the kernel of the representation, i.e., Gal(K/ℚ) = S. Then, by the calculation in (a) above, the proportion of elements of Gal(K/ℚ) whose image lies in S′ is approximately 1/l.

Let C denote the set {Frob_{π}} for primes π ∈ K with ρ̄_{f,λ} × ρ̄_{g,λ}(Frob_{π}) ∈ S′. Then C is clearly invariant under conjugation, and thus we can apply the effective version of the Chebotarev density theorem, Proposition 2.3. We see that |C|/|Gal(K/ℚ)| = |S′|/|S| and thus,

π_{C}(x) ≪ \frac{|S′|}{|S|}π(x)

By property (ii) of the Galois representations ρ_{f,λ} and ρ_{g,λ}, we see that

♯{p ≤ x; a_{p}(f) = a_{p}(g)} ≪ ♯{p ≤ x; Frob_{π} ∈ C}

for some π dividing p.

Thus, we see that

♯\{p ≤ x; a_{p}(f) = a_{p}(g)\} ≪ \frac{|S′|}{|S|}π(x)

if x is sufficiently large.

By property (i) of the representations, we know that K is ramified only at l, and so we can apply Proposition 2.4. Using equation 1, we see that

log d_{L} ≤ 8l^{7} log l.

Thus, for large x, if we choose l ∼ (log x)^{1/8}, we see that the conditions of Proposition 2.3 are satisfied, and so we have, by all of the above,

♯\{p ≤ x; Frob_{π} ∈ C ≪ \frac{1}{l}π(x) ≪ x/(log x)^{9/8}

This proves the theorem.

6. Conditional estimates

The estimates in Theorem 1.1 and Theorem 5.2 can be improved considerably in we assume the generalised Riemann hypothesis for Dedeking Zeta functions of number fields (GRH) ( see [9]). By the methods of [9], it follows that

Theorem 6.1. Assume GRH for Dedekind zeta functions of number fields. If T_{q}(x) is irreducible for some prime q, then

♯{p ≤ x; T_{p}(x) is reducible} ≪ x^{1−δ}

for some δ > 0

and

Theorem 6.2. Assume GRH for Dedekind Zeta functions of number fields. If k_{1} ≠ k_{2}, then

♯{p ≤ x; a_{p}(f) = a_{p}(g)} ≪ x^{1−δ}

for some δ > 0

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[8] Hida, Haruzo; Maeda, Yoshitaka Non-abelian base change for totally real fields. Olga Taussky-Todd: in memoriam. Pacific J. Math. (1997), Special Issue, 189–217.

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[11] Serre, Jean-Pierre Quelques applications du theoreme de densite de Chebotarev. (French) [Some applications of the Chebotarev density theorem] Inst. Hautes Etudes Sci. Publ. Math. No. 54, (1981), 323–401.

[12] Stein, William www.modular.fas.harvard.edu/ stein

E-mail address: sbaba@math.mcgill.ca

Department of mathematics, McGill University, Montréal, H3A 2K6, Canada

E-mail address: murty@mast.queensu.ca

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada