Prime density results for Hecke eigenvalues of a Siegel cusp form

TL;DR

This paper investigates the distribution of Hecke eigenvalues of Siegel cusp forms, providing explicit upper bounds on the density of primes with large eigenvalues, especially for genus 2 forms.

Contribution

It offers explicit bounds on the density of primes with large Hecke eigenvalues for Siegel cusp forms, extending previous results to higher genus and specific cases.

Findings

Explicit upper bounds for prime densities with large eigenvalues

Results specialized for genus 2 Siegel cusp forms

Conditions assuming local temperedness of automorphic representations

Abstract

Let F in S_k(Sp(2g, Z)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues mu_F(n). Suppose that the associated automorphic representation pi_F is locally tempered everywhere. For each c>0 we consider the set of primes p for which |mu_F(p)| >= c and we provide an explicit upper bound on the density of this set. In the case g=2, we also provide an explicit upper bound on the density of the set of primes p for which mu_F(p) >= c.