Course Identification

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

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.

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.

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.)