**189-709B:**Introduction to Automorphic Forms

**Professor:** Henri Darmon

**Classes: ** Tuesday 3:00-5:00.

**Room: ** McGill University, Burnside Hall BH 708.

**Joint Seminar with Goren's course**: Thursday 4:00-6:00
(or when we drop from exhaustion, whichever
comes first).

**Room:** Concordia University, library building LB 559-6.

**Office hours: ** Wednesday 11:30-1:00, in my office (BH 1221).

**Syllabus**:

The course will begin with a discussion of Tate's Thesis, which
amounts to the theory of automorphic forms on GL(1).
Tate's
work was published as a chapter in the famous Proceedings
volume "Algebraic Number Theory"
edited by
Cassels and
Frolich, and I will hand it out in class to those who are unable to
secure a copy.

We will initially follow the book of
Ramakrishnan and Valenza, which is devoted entirely to this topic,
and provides
alot of helpful bacground material. It is
highly recommended for the students who are seeing this material for the
first time.

**Automorphic forms on the web:**

Henri Darmon's web page:
http://www.math.mcgill.ca/~darmon

Virtual study group of Bump's book.

The work of Robert Langlands.

Paul Garrett's reading
list

Vignettes on
automorphic forms, representations, L-functions, and number theory, by
Paul Garrett.

**Textbooks:**

**Compulsory**:

*Daniel Bump*,
Automorphic Forms and Representations, Cambridge University Press.

*John Tate*, Fourier analysis in number fields and Hecke's zeta-function.

*Dinakar Ramakrishnan and Robert Valenza*,
Fourier analysis on number fields.

**Optional (may be used as references)**

Stephen Gelbart,
Automorphic forms on adele groups.

**Extra Reading list:**
The following articles are mostly surveys which I strongly urge you to
read in the course of the semester. They are rather non-technical, and
strive to provide you with the "big picture" and motivation for the topics
that we will be covering in more depth in the lectures.

Robert Langlands,
Representation Theory: its rise and role in number theory, Proceedings of the Gibbs Symposium, 1989.

Robert Langlands,
Where stands functoriality today?, AMS, 1997.

M.R. Murty,
A motivated introduction
to the Langlands program, in Advances in Number Theory, (eds. F. Gouvea
and N. Yui), (1993) 37-66, Clarendon Press, Oxford.

** Grading Scheme **:
Your grade will be based largely on your participation in the Thursday
seminar. A good level of participation will require frequent interventions in
the seminar, and a submission of a written report on
what you cover in the seminar, in the form of files that can be made
available on the web. As the main theme of the seminar we will try to cover
the first chapter of Bump's book on Automorphic forms.