Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$

Abstract

Hyper-Kloosterman sums of different moduli appear naturally in Voronoi's summation formula for cusp forms for $\operatorname{GL}_m(\mathbb{Z})$. In this paper their square moment is evaluated and their bounds are proved in the case of consecutively dividing moduli. As an application, smooth sums of Fourier coefficients of a Maass form for $\operatorname{SL}_m(\mathbb{Z})$ against an exponential function $e(\alpha n)$ are estimated. These sums are proved to have rapid decay when $\alpha$ is a fixed rational number or a transcendental number with approximation exponent $\tau(\alpha) > m$. Non-trivial bounds are proved for these sums when $\tau(\alpha) > (m+1)/2$.