On the Occurrence of Hecke Eigenvalues and a Lacunarity Question of Serre

TL;DR

This paper establishes bounds on the frequency of specific Hecke eigenvalues for automorphic representations over number fields, addressing a question of Serre and extending results to GL(3).

Contribution

It provides new upper bounds on Hecke eigenvalues for GL(n), including a solution to Serre's lacunarity question for GL(2), and extends these bounds to GL(3).

Findings

Bounds on the number of eigenvalues equal to a fixed complex number.

Bounds on eigenvalues with fixed absolute value for GL(2).

Improved bounds for non-dihedral representations.

Abstract

Let \pi be a unitary cuspidal automorphic representation for GL(n) over a number field. We establish upper bounds on the number of Hecke eigenvalues of \pi equal to a fixed complex number. For GL(2), we also determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number \gamma; in the case \gamma=0, this answers a question of Serre. These bounds are then improved upon by restricting to non-dihedral representations. Finally, we obtain analogous bounds for a family of cuspidal automorphic representations for GL(3).