An effective version of the Kuznetsov trace formula for GSp(4)

TL;DR

This paper develops an explicit and effective Kuznetsov trace formula for GSp(4), enabling detailed analysis of automorphic forms, L-functions, and spectral distributions, advancing understanding of the arithmetic and spectral aspects of GSp(4).

Contribution

It provides the first explicit, effective version of the Kuznetsov trace formula for GSp(4), with applications to automorphic forms, L-functions, and zero distribution analysis.

Findings

Weyl law for GSp(4) automorphic forms

Density results on non-tempered spectrum

Bounds on moments and zero distributions of L-functions

Abstract

We develop an explicit version of the Kuznetsov trace formula for GSp(4), relating sums of Fourier coefficients to Kloosterman sums. We study the precise analytic behaviour of both the spectral and the arithmetic transforms arising in the Kuznetsov trace formula for GSp(4). We use these results to provide an effective version of the trace formula, and establish various results on the family of Maa{\ss} automorphic forms on GSp(4) in the spectral aspect: the Weyl law, a density result on the non-tempered spectrum, large sieve inequalities, bounds on the second moment of the spinor and standard $L$-functions, as well as a statement on the distribution of the low-lying zeros of these $L$-functions, determining the associated types of symmetry.