Irreducibility and Galois group of Hecke polynomials

TL;DR

This paper investigates the irreducibility and Galois groups of Hecke polynomials associated with cusp forms, establishing that if one such polynomial is irreducible with a full symmetric Galois group, then this property holds for all primes.

Contribution

It proves a general result linking the irreducibility and Galois group structure of Hecke polynomials across all primes based on a single prime case.

Findings

If T_{n,k}(X) is irreducible with full symmetric Galois group for some n>1, then T_{p,k}(X) shares these properties for all primes p.

The paper establishes a uniform behavior of Galois groups of Hecke polynomials across primes.

Provides new insights into the algebraic properties of Hecke operators and their characteristic 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.