Irreducibility and Galois group of Hecke polynomials

Abstract

Let T_{n,k}(X) be the characteristic polynomial of the n-th Hecke operator acting on the space of cusp forms of weight k for the full modular group. We show that if there exists n>1 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 all primes p.

arXiv commentThere is an error in section 3, in using Theorem 3.2: the function f_{k,Dp^2,d} is not an eigenform a priori