Kloosterman sums on orthogonal groups

TL;DR

This paper investigates Kloosterman sums on specific orthogonal groups, providing explicit descriptions and bounds using algebraic geometry and p-adic analysis techniques.

Contribution

It offers the first explicit descriptions and bounds for Kloosterman sums on $SO_{3,3}$ and $SO_{4,2}$, connecting these sums to multi-dimensional exponential sums.

Findings

Explicit descriptions of Kloosterman sums on the groups.

Bounds established using algebraic geometry and p-adic analysis.

Connections made between sums and multi-dimensional exponential sums.

Abstract

We study Kloosterman sums on the orthogonal groups $SO_{3,3}$ and $SO_{4,2}$, associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and $p$-adic analysis.