We review, and motivate, the notion of a Drinfeld module. A Drinfeld module can be viewed as an analytic analogue of an elliptic curve.
Our main application is a description of explicit class field theory for F_q(t). Explicit class field theory asks for a construction of the maximal abelian extension of F_q(t). In the 1930's, Carlitz used the simplest Drinfeld module to construct a large abelian extension of F_q(t); it is an analogue of the cyclotomic extension of the rationals. In 1974, Hayes built off the work of Carlitz to give an explicit class field theory. I will describe a recent formulation that is not as ad hoc as Hayes.
This talk will be accessible to graduate students.