A weighted vertical Sato-Tate law for Maa{\ss} forms on $\rm{GSp}_4$

TL;DR

This paper establishes a weighted Sato-Tate law for automorphic forms on GSp(4), analyzing the distribution of Satake parameters with respect to varying levels, using advanced trace formulas and local integral calculations.

Contribution

It introduces a weighted Sato-Tate law for GSp(4) automorphic forms, including refinements for squarefree levels and connections to combinatorial problems in the spectral analysis.

Findings

Proved a weighted Sato-Tate law for GSp(4) automorphic forms.

Refined results for the cuspidal spectrum at squarefree levels.

Connected spectral bounds to combinatorial questions involving double cosets.

Abstract

We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on $\rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the $\rm{GSp}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of $\rm{GSp}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P \backslash G / H$, for each parabolic subgroup $P$.