Geometric Gauss Sums and Gross-Koblitz Formula over Function Fields

TL;DR

This paper develops an analog of Gauss sums over function fields in positive characteristic, establishing fundamental properties and a Gross-Koblitz-type formula relating these sums to $v$-adic gamma values.

Contribution

It introduces geometric Gauss sums over function fields, proves their key properties, and derives a novel Gross-Koblitz-type formula linking them to $v$-adic gamma functions.

Findings

Established reflection formula, Stickelberger's theorem, and Hasse-Davenport relations for geometric Gauss sums.

Determined absolute values and signs at infinity of these sums.

Derived a Gross-Koblitz-type formula connecting geometric Gauss sums to $v$-adic gamma values.

Abstract

In this paper, we introduce an analog of Gauss sums over function fields in positive characteristic. We establish several fundamental properties, including reflection formula, Stickelberger's theorem, and Hasse-Davenport relations. In addition, we determine their absolute values and signs at infinity. While these results parallel the classical theory of Gauss sums as well as Thakur's "arithmetic" analogs over function fields, our approach differs completely from both of the preceding cases. Specifically, we first prove a Gross-Koblitz-type formula relating geometric Gauss sums to special $v$-adic gamma values. The properties of geometric Gauss sums then follow from the specializations of this formula together with the functional equations of $v$-adic gamma functions.