# Automorphic forms and representations

This semester the algebra seminar series is on Automorphic forms and representations.

The first two talks on algebraic groups will be on July 26 and 28 from 11:30 to 1 by Maneesh Thakur. Thereafter, the talks will be on Thursdays 11:30 to 1:00.
Automorphic forms on reductive groups are generalizations of modular forms and contribute to understanding representations of such groups. In this seminar series, we will discuss these notions and their interplay with other arithmetic objects. Time permitting, we may take excursions into the "Galois side" aspects of the theory. Initial few lectures will cover basics of algebraic groups and then define the basic objects we propose to study.

### (By Maneesh Thakur) Lecture 1-2:- 26/7 & 28/7

Overview: How automophic/modular forms show up in the study of quadratic forms.
### (By Maneesh Thakur) Lecture 3-5:- 4/8, 11/8 & 18/8

Introduction to algebraic groups.
### (By Maneesh Thakur) Lecture 6-7:- 25/8, 8/9

We will define the notion of Hecke algebras for reductive groups over number fields and introduce the notion of (g,K) modules and their relations with representation theory of G.
### (By Ramdin Mawia) Lecture 8-11:- 15/9, 22/9, 29/9, ..

We will look at automorphic forms on GL(1) (Tate's thesis), followed by automorphic forms and representations on GL(2) (Jacquet-Langlands Theory).

## References

Some of the sources we will be following:

* J.R. Getz, An Introduction to automorphic representations.

* A. W Knapp, Introduction to Langlands' Program.

* S Gelbart, 'An Elementary Introduction to the Langlands Program'