An asymptotic formula of spectral average of central $L$-values on ${\bf GSp}(2)$ for square free levels

TL;DR

This paper develops a new relative trace formula for ${f PGSp}_2$ involving Bessel periods and Rankin-Selberg integrals, leading to weighted equidistribution results for Satake parameters of Siegel cusp forms with square-free levels.

Contribution

It introduces a novel relative trace formula on ${f PGSp}_2$ that connects Bessel periods with spectral data, enabling new equidistribution theorems.

Findings

Weighted equidistribution of Satake parameters for Siegel cusp forms

New relative trace formula involving Bessel periods and Rankin-Selberg integrals

Application to spectral averages of central $L$-values

Abstract

We develop a new kind of relative trace formulas on ${\bf PGSp}_2$ involving the Bessel periods and the Rankin-Selberg type integral a la Piatetski-Shapiro for Siegel cusp forms on its spectral side. As an application, a version of weighted equidistribution theorems for the Satake parameters of Siegel cusp forms of square-free level and of scalar weights is proved.