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189-666A/667B. Graduate Student Seminar

Organiser: Henri Darmon
Time: Fridays, 1:00-3:00 PM, (and sometimes Thursdays).
Room: BH 708.

Ivan Blanco
Luca Candelori
Andrew Fiori
Cameron Franc
Gabriel Gauthier-Shalom
Michael Leahy
Marc Masdeu
Nguyen Ngoc Dong Quan
Luiz Takei
Yu Zhao

This seminar is aimed primarily at my graduate students, although others are welcome to attend and even participate. The participants will lecture on topics that are closely related to, or provide background for, their thesis problems.

Here are a few rules of the seminar:

1. Each week will be devoted to lectures by one or, usually, two students. The lecturer is strongly encouraged to prepare a set of notes and to make these available ahead of their presentation.

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, interruptions from the audience.

4. Participants are encouraged to ask questions during the presentations, at any time.

Here is a tentative list of topics, arranged in rough chronological order. This list includes links to material that you can refer to in order to prepare yourself for the presentations.

Cameron Franc and Marc Masdeu
Differential operators on modular forms, and their geometric interpretation.
Notes written by Cameron and Marc.

Yu Zhao
Modular parametrisations, Heegner points, and an introduction to Kolyvagin's method of ``Euler systems" for modular elliptic curves.
Gross's expository article on this subject

Gabriel Gauthier-Shalom
The Birch-Swinnerton Dyer conjecture for elliptic curves over abelian extensions of Q, and a p-adic variant.

Nguyen Ngoc Dong Quan
An introduction to Sylvester's conjecture.
Article by Dasgupta and VOight

(Time permitting) Minhyong Kim's work on Siegel's theorem.
Kim's paper.

Here is the schedule:

Tuesday, September 2, 1:00 PM.
Organisational meeting, in the Mathematics Lounge.

Friday, September 12, 1:00 PM.
Cameron Franc and Marc Masdeu
A series of lectures devoted to: Algebraic DeRham cohomology. The Gauss-Manin connection. Action of Frobenius on DeRham cohomology. Katz's geometric interpretation of the theta operator on p-adic modular forms.

Friday, September 19, 1:00 PM.
Cameron Franc and Marc Masdeu
Algebraic De Rham cohomology.

Friday, September 26, 1:00 PM.
Cameron Franc and Marc Masdeu
The Gauss-Manin connection.

Gabriel Gauthier-Shalom
An Introduction to the Birch and Swinnerton-Dyer conjecture.

Friday, October 3, 1:00 PM.
Cameron Franc and Marc Masdeu
Calculation of the Gauss-Manin connection: an example.

Nguyen Ngoc Dong Quan
Introduction to the Chabauty method.

Friday, October 10, 1:00 PM.
Cameron Franc
Differential operators acting on modular curves: an overview.

Tuesday, October 14, 1:00 PM.
Yu Zhao
The Shimura-Taniyama conjecture.

Nguyen Ngoc Dong Quan
The Chabauty method, cont'd.

Friday, October 24, 1:00 PM.
Yu Zhao
The Shimura-Taniyama conjecture, cont'd.

Nguyen Ngoc Dong Quan
The Chabauty method, cont'd.

Differential operations on modular forms and the Gauss-Manin connection, cont'd.

Friday, October 31, 1:00 PM.
Yu Zhao
Heegner points, and Kolyvagin's method. (Notes)

Nguyen Ngoc Dong Quan
Syvester's conjecture: an introduction.

Note: Because of Bilu's course we will be meeting on Thursdays in the month of November. All our meetings will be in the 10th floor lounge of Burnside Hall, or in the seminar room at Concordia on seminar afternoons. Exact times are indicated below.

Thursday, November 6, 1:30-3:30 PM.
Yu Zhao
Introduction to Heegner points.

Victoria De Quehen
Modular symbols: from theory to practice.

Thursday, November 13, 9:00-10:00 AM and 4:00-5:00 PM.
AM: Nguyen Ngoc Dong Quan
Sylvester's conjecture (d'après Dasgupta, Voight, ...)

PM: Franc/Masdeu
Frobenius and deRham cohomology.

Thursday, November 20, 10:00-12:00 PM.
Nguyen Ngoc Dong Quan
Sylvester's conjecture, cont'd.

Gabriel Gauthier Shalom
The BSD conjecture over cyclotomic extensions.

Thursday, November 27, 9:00-10:00 AM and 4:00-5:00 PM.
AM: Seminar cancelled, because of the lecture by Alina Bucur, which you should all attend. (see the web page of the qvnts).

PM: Yu Zhao
Kolyvagin's theorem on modular elliptic curves.

Thursday, December 4, 9:00-10:00 AM and 4:00-5:00 PM.
AM: Masdeu/Franc

PM: Zhao, cont'd

Winter semester

Our efforts so far have focussed on reading Bruinier's article on Hilbert modular forms, and related background, in preparation for the Barbados workshop.

Currently, we are learning about the Weil representation and theta lifts.

Here are Luca's notes on Weil's representations, here is the second part, and here is the reference he's been using.