An effective version of the Kuznetsov trace formula for GSp(4)
Félicien Comtat, Didier Lesesvre, Siu Hang Man
TL;DR
The paper develops an explicit Kuznetsov trace formula for $\operatorname{GSp}(4)$, connecting sums of Fourier coefficients to Kloosterman sums via explicitly controlled spectral and arithmetic transforms. It builds a robust analytic framework using a fourfold Mellin–Barnes representation of Whittaker functions to localise the spectral transform and a careful analysis of the arithmetic side to bound Kloosterman-type integrals. This yields concrete consequences: a Weyl law with Plancherel-type density, a density bound for non-tempered spectrum, large sieve inequalities for Fourier coefficients, second-moment bounds for both spinor and standard $L$-functions, quantitative quasi-orthogonality, and one-level density results identifying orthogonal versus symplectic symmetry for Spin and Std $L$-functions, respectively. The results enable arithmetic statistics for GSp(4) in the spectral aspect and provide tools for understanding symmetry types in families of automorphic $L$-functions on this group, with potential applications to subconvexity and distribution of low-lying zeros.
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 Maaß 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.