# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

Tuesday, 13:15 - 15:00, Ziskind, Rm 155

## Field of Study, Course Type and Credit Points

## Comments

## Prerequisites

Basic notions of Functions of Complex variables, of Algebraic Geometry, of Representation Theory and of Number Theory.

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**Take-home exam**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

Course description.

This is a rst part of a year long course on the theory of Automorphic

forms and Automorphic Representations. This theory is now one of the focal

points of Mathematics. It has very many applications in dierent areas of

Number Theory, Physics, Combinatorics and so on.

This theory is in the process of constant development and it is not easy

to formulate what are the goals of the theory. I will try to describe basic

structures of the theory and main ideas developed in it. I will also try to

illustrate them by examples of applications.

I will try to describe the development of this theory in quasi-historical

manner, starting with the classical theory of modular forms.

In Spring semester 2018 I will mostly discuss the classical theory of mod-

ular functions. I will also describe how to pass to the contemporary ap-

proach to automorphic forms based on automorphic representations of adelic

groups.

I hope to continue this course in Fall of 2018.

I am planning to outline main conjectural structures of the theory evolv-

ing around Langlands' program, Generalized Riemann Hypothesis and Ra-

manujan conjecture.

One should realize that though in many cases we can formulate rather

precise conjectures describing the behavior of L-functions and automorphic

representations what we can prove about these conjectures is just scratching

the surface.

Syllabus.

1. Classical theory of modular forms for the group SL(2;Z).

Relation with elliptic curves, Eisenstein series, Structure of the algebra of

modular forms.

Fourier expansion of modular forms.

Dierent constructions of modular forms.

Discriminant function and cusp forms. Properties of coecients (n).

Digression on Dirichlet L-functions.

Hecke operators and Hecke L-functions.

Peterson scalar product.

2. Maass forms and related L-functions.

3. Modular functions with respect to congruence subgroups. New forms

and their L-functions. Weil's inverse theorem.

4. Modular forms with respect to co-compact subgroups. Statement of

the Jacquet-Langlands correspondence.

6. Introduction to Hilbert modular forms.

7. Introduction to Siegel modular forms.

8. Automorphic forms and Representation Theory.

Digression on representations of the group SL(2;R). Interpretation of

automorphic forms in terms of automorphic representations.

9. Adelic automorphic representations.

Adeles.

Tate's thesis and L-functions

Passing to adelic automorphic representations.

Hecke L-function of an automorphic representation.

10. Langlands' L-function. Class eld theory and Langlands program.

11. Langlands' dual group. Langlands program and functoriality conjec-

ture.

12. Relation of Langlands program to Artin and Ramanujan conjectures.

## Learning Outcomes

The students will learn the classical theory of modular forms and the modern theory of automorphic forms and automorphic representations.

They will understand the connections to number theory and representation theory.

Thus, they will gain access to the ample literature on the subject and will be able to identify the relation to automorphic forms in their future research in pure mathematics or theoretical physics.

## Reading List

1. "1-2-3 of modular forms" - Lectures at a Summer School in Nordfjordeid, Norway

Authors: **Bruinier**, J.H., **van der Geer**, G., **Harder**, G., **Zagier**, D.

Editors: **Ranestad**, Kristian (Ed.)

2. "An Introduction to the Langlands Program"

Editors: **Bernstein**, Joseph, **Gelbart**, Steve (Eds.)