Drinfeld Modules

Abstract

Drinfeld curves have appeared in the study of codes and the associated geometry in the book of Tsfasman and Vladut (1991), and in Manin and Vladut (1984). Niederreiter and Xing specifically have studied Drinfeld modules of rank one in their book and recent papers; v., Xing and Niederreiter (1996, 1999) and Niederreiter and Xing (1997, 2001). In this chapter the focus is on the analogies in results between elliptic curves and Drinfeld curves, in particular in the area of LangTrotter conjectures, the Riemann hypothesis, the Hasse-Weil inequality, zeta functions and the results of Gekeler (2001) and Schweizer (2000). The results of Elkies and Gekeler on Drinfeld towers are presented. In Chapter 7, works of Scanlon (2001) and Gillard et al. (2003) regarding Drinfeld modules and cryptography are discussed. Other applications of Drinfeld modules are discussed by Panchishkin (1992).