Drinfeld Modules: An Introduction

Abstract

In the last chapter we introduced a special class of Drinfeld modules for the ring A = F[T] defined over the field k = F(T) and discussed some of their properties. By considering the Carlitz module, in particular, we were able to construct a family of field extensions of k with properties remarkably similar to those of cyclotomic fields. In this chapter we will give a more general definition of a Drinfeld module. The definition and theory of these modules was given by V. Drinfeld in the mid-seventies, see Drinfeld [1, 2]. The application of the rank 1 theory to the class field theory of global function fields is due to Drinfeld and independently to D. Hayes [2]. The article by Hayes [6] provides a compact introduction to this material. A comprehensive treatment of Drinfeld modules (and, even more generally, T-modules) can be found in the treatise of Goss [4].