## McGill University

# Department of Mathematics & Statistics

# Introduction to Automorphic Forms

# 189-709B

## Detailed Syllabus

**Tuesday, January 15**. Topological groups.

**Thursday, January 17**. Antoine Gournay. The Riemann zeta-function and its
analytic continuation. (Following Bump's book.)

(dvi,
ps,
pdf.)

**Tuesday, January 22**. Representations of locally compact groups:
preliminaries and generalities. Banach spaces and Banach
algebras. The
Gelfand transform.

**Thursday, January 24.** Eyal Goren and Matthew Greenberg.
Introduction to Riemann surfaces.

**Tuesday, January 29.** Representations of locally compact groups:
Spectral theorems. Unitary representations.

**Thursday, January 31.** We will be in Vermont on that day...

**Suggested Exercises**: Chapter 2, exercises 9, 10, 16, 17, 18, 19,
20, 21, 23-26, 28.

**Tuesday, February 5.** The Pontryagin dual. Functions of positive type.

**Thursday, February 7.**

Matthew Greenberg: Introduction to
Riemann surfaces (cont'd).

Natalia Archinard: Riemann surfaces.

**Suggested Exercises**: Chapter 3, exercises 3, 4, 5.

**Tuesday, February 12.**
Fourier inversion formula.
Pontryagin duality.

**Thursday, February 14.**

Melissande Fortin-Boisvert: The Riemann Hurwitz formula.

**Suggested Exercises**: Chapter 3, exercises 7, 8, 13, 14.

**Tuesday, February 19.** ** Spring Break**. Class is cancelled.

**Thursday, February 21.**

**Tuesday, February 26**:
Fourier inversion formula.
Pontryagin duality.

**Thursday, February 28**

Cristina Toropu: Dirichlet characters and the analytic continuation
of Dirichlet L-series.

(dvi,
ps,
pdf.)

**Tuesday, March 5**:
* Class is cancelled *

**Thursday, March 7**

I will lecture on chapter 4 of Ramakrishnan-Valenza: Arithmetic fields,
locally compact fields, global fields.

**Suggested Exercises**: Chapter 4, exercise 2, 4, 5, 6, 7, 10, 13, 15.

**Tuesday, March 12**:
Chapter 5 of Ramakrishnan-Valenza: Adeles, Ideles and the class group.

**Thursday, March 14**

**March-Hubert Nicole**: Section 1.2 of Bump's book (the modular group),
including exercises 1.2.7 to 1.2.11.

(dvi,
ps,
pdf.)

**Suggested Exercises**: Chapter 5, exercise 1, 3, 4, 6, 10, 15.

**Tuesday, March 19**:
End of Chapter 5 of Ramakrishnan-Valenza: Adeles, Ideles and the class group.

**Thursday, March 21**

**Antoine Arbour**: Completing unramified coverings
of Riemann surfaces
to branched coverings of compact Riemann surfaces. Application to Galois
coverings.

**Tuesday, March 26**:
Chapter 7.1 of Ramakrishnan-Valenza: Local zeta functions and the local
calculations in Tate's thesis.

**Suggested Exercises**: Chapter 7, Exercises 1 and 3.

**Thursday, March 28**

** Vasilisa Chramtchenko**: Linear differential equations over a Riemann surface, I.

**Maxim Samsonov**: Linear differential equations over a Riemann
surface, II.

**Tuesday, April 2**:
Chapter 7.2 of Ramakrishnan-Valenza: The adelic Poisson summation formula, and
the Riemann-Roch theorem.

**Suggested Exercises**: Chapter 7, Exercises 4, 5, 6, 12.

**Thursday, April 4**

**Maxim Samsonov**: Linear differential equations over a Riemann
surface, III.

** Marc-Hubert Nicole**: Triviality of vector bundles over a non-compact
Riemann surface.

**Tuesday, April 9**:
Chapter 7.3 and 7.4 of Ramakrishnan-Valenza: the
global functional equation for Hecke L-series.

** **