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.

Notes written by Cameron and Marc.

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

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

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:

Organisational meeting, in the Mathematics Lounge.

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.

Algebraic De Rham cohomology.

The Gauss-Manin connection.

An Introduction to the Birch and Swinnerton-Dyer conjecture.

Calculation of the Gauss-Manin connection: an example.

Introduction to the Chabauty method.

Differential operators acting on modular curves: an overview.

The Shimura-Taniyama conjecture.

The Chabauty method, cont'd.

The Shimura-Taniyama conjecture, cont'd.

The Chabauty method, cont'd.

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

Heegner points, and Kolyvagin's method. (Notes)

Syvester's conjecture: an introduction.

Introduction to Heegner points.

Modular symbols: from theory to practice.

Sylvester's conjecture (d'après Dasgupta, Voight, ...)

Frobenius and deRham cohomology.

Sylvester's conjecture, cont'd.

The BSD conjecture over cyclotomic extensions.

TBA

Kolyvagin's theorem on modular elliptic curves.

TBA

TBA

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.