Kloosterman sums on orthogonal groups

TL;DR

This paper analyzes mathematical structures called Kloosterman sums in specific orthogonal groups using advanced algebraic and p-adic methods.

Contribution

The paper provides explicit descriptions and bounds for Kloosterman sums associated with specific Weyl group elements in orthogonal groups.

Findings

Kloosterman sums on SO₃,₃ and SO₄,₂ are described using multi-dimensional exponential sums.

Bounds for these sums are derived using algebraic geometry and p-adic analysis.

Abstract

We study Kloosterman sums on the orthogonal groups \documentclass[12pt]{minimal}
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\begin{document}$$SO_{3,3}$$\end{document}SO3,3 and \documentclass[12pt]{minimal}
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\begin{document}$$SO_{4,2}$$\end{document}SO4,2, 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.