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.