## Math 9309B: Automorphic Forms, Winter 2015

#### Course Outline:

We shall study two types of L-functions in this course: arithmetic L-functions andautomorphic L -functions. A main theme is the relation between the two, and questions of analytic continuation, functional equation, and Euler product formula.

Some of the most famous examples of arithmetic L-functions are Riemann's zeta function and the Dirichlet L-functions. Simplest types of automorphic L-functions are those defined by the Fourier expansion of modular forms.

An example of an automorphic form is a classical modular form. A modular form, in its simplest type, is a holomorphic function on the upper half plane that is quasi-periodic with respect to the action of the integral modular group . A crucial observation is that if we lift a modular form to the group SL_2 (R) we obtain a function that is closely tied to the representation theory of the latter group. This makes it possible to define similar analytic objects for all Lie groups, and hence define automorphic forms for Lie groups. For applications to number theory, it is crucial to work with adelic Lie groups and their representation theory. Among other things, we shall verify that the L -function defined via adelic GL(2) coincide with L-functions defined through modular forms. We shall also look at number theoretic consequences of such statements.

This course is a gentle introduction to the theory of automorphic forms. The following topics will be covered:

1. Elliptic functions,

2. Modular forms for SL_2(Z),

3. Representation theory of SL_2(R),

4. P-adic numbers and Fourier analysis on Adeles and Ideles,

5. Tate's thesis,

6. Automorphic representations of GL_2 (A),

7. Automorphic L-functions.

#### Student presentations:

.

#### References:

1. I shall closely follow: Automorphic Forms by A. Deitmar. Library has an online copy.2. Automorphic forms and the Langlands program (International press). Survey articles in this volume by Knapp, Gelbart, Gan, and Lapid are recommended.