Gamma Functions for Function Fields and Drinfeld Modules

Abstract

The purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field. In particular, we explore, among other things, their connection with the theory of Drinfeld modules of rank one (which is function field cyclotomic theory), their interpolations, functional equations, the nature of their special values (transcendence, algebraicity, connection with periods) and various analogies. Most of the facts about the classical gamma function are recalled when necessary. Readers may refer to [G-K] or Artin's book on the classical gamma function. Now we describe more precisely the contents of this paper. Let Fq be a finite field of characteristic p. Section 0 covers the background material on Drinfeld modules. It also describes the gamma function F for Fq[T] (with