On Sums of hyper-Kloosterman Sums

TL;DR

This paper extends the spectral interpretation of Kloosterman sums to hyper-Kloosterman sums using $GL(3)$ automorphic forms, resolving a longstanding problem and introducing new bounds for derivatives of the $J$-Bessel function.

Contribution

It constructs a new formula for hyper-Kloosterman sums via $GL(3)$ automorphic forms, advancing the understanding of their spectral theory.

Findings

Derived a spectral formula for hyper-Kloosterman sums

Established new bounds for derivatives of the $J$-Bessel function

Discussed the original method of Bump, Friedberg, and Goldfeld

Abstract

A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of $GL(2)$ automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using $GL(3)$ automorphic forms, resolving a long-standing problem of Bump, Friedberg and Goldfeld. Along the way, we develop what are apparently new bounds for the order derivatives of the classical $J$-Bessel function, and we conclude with a discussion of the original method of Bump, Friedberg and Goldfeld.