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).

January 16.

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.

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

January 30 and February 5.

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.

Hecke operators (Chapter 2 of Serre's article)

Febrary 19.

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

February 26.

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

March 5.

No seminar.

March 12.

March 19.

Classical modular forms of level

March 26.

Introduction to the ``algebraic" point of view on modular forms.

April 2.

The Tate curve and the q-expansion principle.

April 9.

April 16.

April 23.