Geometric Gauss Sums and Gross-Koblitz Formula over Function Fields

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.