Topics in Number Theory

This course will cover the basics of automorphic forms on Lie groups. I will mainly stick to Chevalley groups for simplicity, and begin the course by summarizing the simplest example of the subgroup SL(2,Z) of SL(2,R), along with its modular forms. This has been the topic of a number of recent graduate courses, and while none of these are formally a prerequisite for the material in this class, they certainly motivate the material of this course. The goal is to give students background to work with these objects on more general groups, which have become crucial to many recent advances in analytic number theory. Additionally, arithmetic subgroups of Lie groups are important in group theory, dynamics, and topology.

I will assume no particular knowledge of Lie groups, and instead build up from scratch. My plan is to cover some of the following topics, stressing the SL(2) example as motivation. However, alterations to this plan may be made depending on the interests of the audience.

1. Overview of SL(2,Z) as a discrete subgroup of SL(2,R), and its automorphic forms.

2. The notion of algebraic group, and some key features, such as Borel subgroups.

3. Root systems and Weyl groups.

4. Cartan-Killing classifications of Lie groups and symmetric spaces.

5. Chevalley groups, Steinberg and Serre relations

6. Finite dimensional representations, and theory of highest weight.

7. Arithmetic subgroups, examples and constructions. Rigidity Theorems.

8. Reduction theory and the LLL algorithm.

9. Adelization

10. Hecke operators

11. Differential operators, such as the Laplacian

12. Automorphic representations

13. Ramanujan conjecture

14. Selberg Laplace eigenvalue conjecture

15. Langlands' Functoriality conjectures