1. Background

Function Field Arithmetic is important and highly developing area in Number Theory and Arithmetic Geometry. Two recent fascinating breakthroughs are Vincent Lafforge’s new proof of the Langlands correspondence, and Zhiwei Yun and Wei Zhang’s work of the Gross-Zagier formula, both over function fields. The (modest) purpose of this course is to introduce graduate students as well as motivated senior undergraduates the prerequisites (or the first basic accounts) into this area so that more young can participate and possibly find problems inspiring from the development number fields or modular curves etc, to work with. We plan to introduce Drinfeld modules, Drinfeld modular varieties, and some of t-modules as long as time permits.

2. Outline

We will mainly follow Goss [1] Chap. I, IV and V.

Outline of contents of this course:

1.Additive polynomial, Moore determinants, left and right division, finite additive subgroups of G_a.

2. Carlitz and Drinfeld modules, analytic theory: lattices in C_infty and Drinfeld modules over C_infty, height, rank, morphism and rigilidy, action by ideals and tensor with ideals, Drinfeld modules over finite fields, families of Drinfeld modules and their mouli spaces.

3. Vector bundles and connections, varphi-sheaves, t-modules, pure t-modules, tensor power of Carlitz modules.

3. Credit: 3

4. References: 

[1] D. Gross, Basic Structures of Function Field Arithmetic. (1996)

[2] V. Drinfeld, Elliptic modules, (English translation) Math. USSR-Sb. 23 (1976), 561--592.

[3] G. Laumon, Cohomology of Drinfeld modular varieties. Part I, (1996).