Bounds for Kloosterman sums on GL(n)

TL;DR

This paper derives power-saving bounds for Kloosterman sums on GL(n), providing explicit exponential sum representations and advancing understanding beyond existing conjectures, including bounds for GL(4).

Contribution

It introduces explicit exponential sum representations for Kloosterman sums on GL(n) and extends bounds beyond Sarnak's density conjecture, including for GL(4).

Findings

Established power-saving bounds for Kloosterman sums on GL(n).

Provided explicit exponential sum representations.

Extended bounds beyond existing conjectures, including for GL(4).

Abstract

This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for GL(n), as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as exponential sums. As an application we go beyond Sarnak's density conjecture for the principal congruence subgroup of prime level. We also bound all Kloosterman sums for GL(4).