On the distribution of Hecke eigenvalues for cuspidal automorphic representations for GL(2)

TL;DR

This paper proves the existence of infinitely many large positive and negative Hecke eigenvalues for self-dual cuspidal automorphic representations of GL(2), addressing a question posed by Serre and extending to non-self-dual cases.

Contribution

It establishes the distribution of Hecke eigenvalues for GL(2) automorphic representations, including explicit bounds and non-self-dual cases, advancing understanding in automorphic forms.

Findings

Infinitely many Hecke eigenvalues exceed a positive constant.

Infinitely many Hecke eigenvalues are below a negative constant.

Results answer a question of Serre regarding eigenvalue distribution.

Abstract

Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke eigenvalues that are less than an explicit negative constant. This provides an answer to a question of Serre. We also consider analogous problems for cuspidal automorphic representations that are not self-dual.