Gauss sums forF _q [T]

Abstract

The purpose of this paper is to define ‘Gauss sums’ taking values in function fields of one variable over a finite field and to prove analogues of various classical and recent results. These results include Stickelberger's theorem, the Hasse-Davenport theorem, Weil's theorem on ‘Jacobi sums as Hecke characters’ and the Gross-Koblitz theorem. For comparison, the reader may consult [G-K] and references given there.

In this paper we deal only with the simplest case, where the base ringA is the polynomial ringF _q [T] and where we use the Carlitz module; i.e., the simplest rank one Drinfeld module. (See section I). The general case, which has a quite different flavour, will be presented elsewhere. These results formed a part of the author's thesis, ‘Gamma functions and Gauss sums for function fields and periods of Drinfeld modules’ (Harvard 1987). But the new presentation here is due to a suggestion by Professor Tate. It is my pleasure to thank him.