Symplectic Kloosterman Sums and Poincaré Series

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.