Local spectral equidistribution for Siegel modular forms and applications

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 Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

45 pages; typos corrected and two references added; version to appear in Compositio Math

References in corpus (1)

• A note on Fourier coefficients of Poincaré series

Cited by in corpus (17)

• Local spectral equidistribution for degree 2 Siegel modular forms in level and weight aspects
• On the distribution of Satake parameters for Siegel modular forms
• An equidistribution theorem for holomorphic Siegel modular forms for $GS{p}_4$
• A weighted one-level density of families of $L$-functions
• Maass relations for Saito-Kurokawa lifts of higher levels
• Weighted one-level density of low-lying zeros of Dirichlet $L$-functions
• Weighted distribution of low-lying zeros of GL(2) L-functions
• Local and global Maass relations (expanded version)
• Sato-Tate theorem for families and low-lying zeros of automorphic $L$-functions
• Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect
• Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2) II
• Weighted low-lying zeros of L-functions attached to Siegel modular forms
• A relation between multiplicity one and Bocherer's conjecture
• Spectral summation formula for GSp(4) and moments of spinor L-functions
• Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values
• The Ranking-Selberg integral on $\mathbf{GSp}(2)$ for square free levels
• Linear Relations of Siegel Poincare Series and Nonvanshing of the Central Value of Spinor L-functions