Symplectic Kloosterman Sums and Poincar\'e Series

TL;DR

This paper establishes power-saving bounds for Kloosterman sums on Sp(4) using stratification and p-adic stationary phase, linking these sums to Fourier coefficients of Poincaré series.

Contribution

It introduces new bounds for Sp(4) Kloosterman sums and connects them to Fourier coefficients of Poincaré series, advancing understanding of automorphic forms.

Findings

Proved power-saving bounds for Kloosterman sums on Sp(4).

Established a relation between Kloosterman sums and Fourier coefficients.

Applied stratification and p-adic stationary phase methods.

Abstract

We prove power-saving bounds for general Kloosterman sums on $\operatorname{Sp}(4)$ associated to all Weyl elements via a stratification argument coupled with $p$-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of $\operatorname{Sp}(4)$ Poincar\'e series.