On the irreducibility of Hecke polynomials

Abstract

Abstract. Let T n,k (X) be the characteristic polynomial of the nth Hecke operator acting on the space of cusp forms of weight k for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials T n,k (X). Using this criterion with some machine computation, we show that if there exists n ≥ 2 such that T n,k (X) is irreducible and has the full symmetric group as Galois group, then the same is true of T p,k (X) for each prime p ≤ 4, 000, 000.

Constancy of adjoint L-invariant

“…In the 1970s, Y. Maeda made a conjecture asserting that Q(a(p, f )) for any normalized Hecke eigenform f in S k (SL 2 (Z)) has degree equal to d := dim S k (SL 2 (Z)) with its Galois closure having Galois group isomorphic to the symmetric group S d of d letters. This conjecture is numerically verified up to large weight and large p (e.g., [1]) and implies our conjecture if N = 1.…”

“…In the above proof, the two conditions Φ(1, 1) = 0 and geometric irreducibility are both indispensable. If we do not suppose Φ(1, 1) …”

Divisibility of an eigenform by an eigenform

“…Kleinerman [13] showed that T 2,k (x) is irreducible up to weight 3000. Alhgren [1] showed for all weights k that if for some n, T n,k (x) is irreducible and has a full Galois group, then T p,k (x) does as well for all p ≤ 4, 000, 000. Finally from correspondence between Stein and Ghitza it is known that T 2,k (x) is irreducible up to weight 4096.…”

Untitled

“…Let k be a positive integer, and let l be a prime number. The Hecke polynomial of T Some progress has been made towards this conjecture; methods introduced in [4] prove that certain Hecke polynomials are irreducible and have full Galois group, and results such as those in [6], [2] and [1] show that if a certain T l is irreducible then other T r must be irreducible also.…”

“…In [7] and [2], the authors first show that certain of the T 1,1 2,k are irreducible and then using effective versions of the Chebotarev density theorem to show that a positive proportion of the T 1,1 l,k are irreducible, and in [1], plausible hypotheses on the non-vanishing of certain coefficients of modular forms are used to prove irreducibility of the Hecke polynomials (when the weight is 12, for instance, these hypotheses are exactly Lehmer's conjecture on the Ramanujan τ -function).…”