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189-666A/667B. Graduate Student Seminar

Organiser: Henri Darmon
Time: Tuesday 2:00-4:00.
Room: ***** NOTE THE CHANGE **** BH 920.

Regular Participants:
Dylan Attwell-Duval
Francesca Bergamaschi
Miljan Brakocevic
Luca Candelori
Francesc Castella
Amy Cheung
Victoria de Quehen
Yara Elias
Andrew Fiori
Michele Fornea
Shan Gao
Jochen Gartner
Clément Gomez
Bruno Joyal
Antonio Lei
Bahare Mirza Hossein
Giulio Orrechia
Juan Ignacio Restrepo
Nicolas Simard
Luiz Takei
Maxime Turgeon
Chih Yun Chuang

This seminar is aimed at all graduate students in the number theory group. This year's theme is the theory of modular forms and p-adic modular forms. The ultimate (perhaps overly ambitious) goal is to understand the Buzzard-Taylor technique of p-adic analytic continuation of overconvergent forms, and its recent extension by Kassaei to totally real fields. We may not get to that last part, but at least the seminar should leave you well equipped at the end to understand the work of Kassaei.

Our main references, before we get to that, will be:

1. Serre's foundational paper on p-adic modular forms.

2. Katz's equally foundational paper on p-adic modular forms and their geometric interpretation.

Here are a few rules of the seminar:

1. Each week will be devoted to lectures by one student.

2. This is a working seminar, aimed at students with varying backgrounds. It is important that lectures be accessible to all participants.

3. Speakers should allow, in fact, welcome, questions, interruptions, and constructive comments from the audience.

4. Participants are encouraged to ask questions during the presentations, at any time, and to put in their two cents' worth.

Here is the schedule (to be made up as we go along).

Winter Semester Schedule

January 16. Michele Fornea.
Introduction to modular forms. Proof that the space of modular forms of a given weight on SL(2,Z) is finite dimensional, and that the graded ring of modular forms is generated by the Eisenstein series of weights 4 and 6.

January 23. Giulio Orrechia
An overview of the theory of modular forms mod p, following Serre and Swinnerton-Dyer.

January 30 and February 5. Francesca Bergamaschi.
p-adic modular forms. The motivating problem: construction of p-adic L-functions for totally real fields. Francesca's notes are posted here.

February 12. Juan Restrepo
Hecke operators (Chapter 2 of Serre's article)

Febrary 19. Michele Fornea
Iwasawa functions as constant terms of Eisenstein series. (Chapter 4 of Serre's article.)

February 26. Michele Fornea
Iwasawa functions as constant terms of Eisenstein series. (Chapter 4 of Serre's article.)

March 5. McGill Spring break
No seminar.

March 12.
p-adic zeta functions for totally real fields.

March 19. Francesc Castella
Classical modular forms of level p are p-adically of level 1. (Chapter 3 of Serre's article).

March 26. Juan Restrepo
Introduction to the ``algebraic" point of view on modular forms.

April 2. Juan Restrepo
The Tate curve and the q-expansion principle.

April 9.

April 16.

April 23.