A RAPID INTRODUCTION TO DRINFELD MODULES, t-MODULES, AND t-MOTIVES

The theory of Drinfeld modules was initially developed to transport the classical ideas of lattices and exponential functions to the function field setting in positive characteristic. This impulse manifested itself in work of L. Carlitz [9], who investigated explicit class field theory for Fq(θ) and defined the later named Carlitz module, which serves as the analogue of the multiplicative group Gm. The theory was brought to rapid fruition by V. G. Drinfeld [20], who independently superseded Carlitz’s work and further extended the theory to higher rank lattices in his investigation of elliptic modules, now called Drinfeld modules. G. W. Anderson [1] saw correctly how to develop the theory of higher dimensional Drinfeld modules, called t-modules, and at the same time produced a robust motivic interpretation in his theory of t-motives. The present article aims to provide a brief account of the theories of Drinfeld modules and Anderson’s t-modules and t-motives. As such the article is not meant to be comprehensive, but we have endeavored to summarize aspects of the theory that are of current interest and to include a number of examples. For further information and more complete details, readers are encouraged to consult the excellent surveys [17, 32, 39, 50, 56].