Symplectic Kloosterman Sums for $\operatorname{Sp}(2n)$ with Powerful Moduli

TL;DR

This paper establishes a new bound for symplectic Kloosterman sums associated with Siegel modular forms, extending understanding of their behavior for composite moduli and applying results to equidistribution problems.

Contribution

It introduces a non-trivial bound for symplectic Kloosterman sums with composite moduli not divisible by a prime, advancing the analytic theory of automorphic forms on $ ext{Sp}(2n)$.

Findings

Derived a new bound for symplectic Kloosterman sums

Applied bounds to prove equidistribution of coprime symmetric pairs

Extended analysis to non-prime power moduli

Abstract

We prove a non-trivial bound for $\operatorname{Sp}(2n)$ Kloosterman sums of moduli not equal to a prime multiple of the identity. These sums are attached to Siegel modular forms on the group $\operatorname{Sp}(2n)$ and appear in the corresponding Petersson formula. We give an application to equidistribution of coprime symmetric pairs.