Hyper-Kloosterman sums of different moduli and applications

Hyper -Kloosterman sums of different moduli appear naturally in Voronoi's summation formula for cusp forms for $GL_{m}(\mathbb Z)$. In this talk, we give estimates of Hyper-Kloosterman sum in the case of consecutively dividing moduli. As an application, smooth sums of Fourier coefficients of Maass forms for $SL_{m}(\mathbb Z)$ 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 $\tau(\alpha) > (m + 1)/2$.