Local spectral equidistribution for Siegel modular forms and applications

TL;DR

This paper establishes a quantitative local equidistribution result for the Satake parameters of Siegel cusp forms of genus 2 with growing weight, providing evidence for conjectures about Fourier coefficients and zeros of associated L-functions.

Contribution

It introduces a new approach to analyze the distribution of local components of Siegel cusp forms, combining Bessel models and a variant of Petersson's formula for quantitative results.

Findings

Quantitative local equidistribution of Satake parameters

Evidence supporting B"ocherer's conjecture on Fourier coefficients

Analysis of low-lying zeros of spinor L-functions

Abstract

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of B\"ocherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.