On the occurrence of large positive Hecke eigenvalues for GL(2)

TL;DR

This paper proves that for certain automorphic representations of GL(2), there is a positive density of primes where the Hecke eigenvalues exceed a fixed positive threshold, indicating frequent large eigenvalues.

Contribution

It establishes the existence of a positive upper Dirichlet density of primes with large Hecke eigenvalues for self-dual cuspidal automorphic representations of GL(2).

Findings

Positive density of primes with large Hecke eigenvalues

Quantitative bounds on eigenvalue size

Frequency of large eigenvalues among primes

Abstract

Let $\pi$ be a cuspidal automorphic representation for GL(2)/$\mathbb{Q}$ that is self-dual. In this Note we show that there exists a positive upper Dirichlet density of primes at which the associated Hecke eigenvalues of $\pi$ are larger than a specified positive constant.