On the distribution of Satake parameters for Siegel modular forms

TL;DR

This paper establishes a harmonic equidistribution result for Satake parameters of Siegel modular forms, using a new Petersson formula to analyze automorphic representations as level grows.

Contribution

It introduces a novel asymptotic Petersson formula for GSp(2n) in the level aspect, enabling the study of Satake parameter distribution.

Findings

Proves equidistribution of Satake parameters for fixed weight as level increases.

Develops a new asymptotic Petersson formula for GSp(2n).

Provides insights into automorphic representations of PGSp(2n).

Abstract

We prove a harmonically weighted equidistribution result for the $p$-th Satake parameters of the family of automorphic cuspidal representations of $\operatorname{PGSp}(2n)$ of fixed weight $\mathtt{k}$ and prime-to-$p$ level $N\to \infty$. The main tool is a new asymptotic Petersson formula for $\operatorname{GSp}(2n)$ in the level aspect.