On the Lang--Trotter conjecture for Siegel modular forms

TL;DR

This paper proves an adelic open image theorem for genus two Siegel modular forms and investigates the distribution of their Hecke eigenvalues, providing bounds related to the Lang--Trotter conjecture.

Contribution

It generalizes known results to Siegel modular forms and offers new bounds on eigenvalue distributions in this context.

Findings

Proved an adelic open image theorem for Siegel modular forms.

Derived upper bounds for the frequency of specific Hecke eigenvalues.

Extended Lang--Trotter type conjectures to genus two Siegel modular forms.

Abstract

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p \le x : a_p = a\}$ for fixed $a\in\mathbf{C}$, in the spirit of the Lang--Trotter conjecture for elliptic curves.