Advanced Courses on Modularity
 

An activity of an i-MATH Intensive Research Programme

Group Picture NEW!
 
Prof. Berger Notes NEW!
 
Prof. Diamond Notes
 
Prof. Wintenberger Notes
 
List of participants with lodging arranged through the CRM
 
List of participants
 
Programme
   
Dates:   June 14 to 25, 2010
 
Place:   Centre de Recerca Matemàtica (CRM), Bellaterra, Barcelona, Spain          
How to reach the CRM

Subject Matter

Since the work of Wiles and Taylor-Wiles that led to the proof of Fermat's Last Theorem and the Taniyama-Shimura-Weil conjecture several other important Modularity results have been proved, results giving the connection between certain algebraic or geometric objects (such as Galois representations, GL_2 type abelian varieties or rigid Calabi-Yau threefolds) and modular or automorphic forms.
Recent results include very strong Modularity Lifting Theorems in the GL_2 case (in particular the results of Kisin), many of them valid also over totally real number fields, as well as Modularity Lifting Theorems for higher-dimensional self-dual Galois representations.
The role of Mazur's theory of Deformations of Galois representations is crucial in all these developments, and the same applies to the theory of p-adic Galois representations and Fontaine rings.
Together with the powerful Modularity Lifting Theorems (M.L.T.), the Potential Modularity results obtained first by R. Taylor in the 2-dimensional case and then by Taylor, Harris and collaborators in higher dimensions, are also of key importance. With these tools Taylor, Harris and collaborators have been able to give a proof of the Sato-Tate conjecture for elliptic curves over Q.
The Potential Modularity result was also crucial in the work of Khare, Wintenberger and Dieulefait allowing them to achieve (together with Galois deformation and Group representation theories) the key results of ``existence of strongly compatible systems" and ``existence of lifts with prescribed properties", as well as the proof of the first cases of the Fontaine-Mazur and Serre's conjectures. These results combined with M.L.T. and a sophisticated inductive argument led Khare and Wintenberger to a proof of Serre's modularity conjecture in full generality.
In the last years, starting with a paper by Buzzard, Diamond and Jarvis, a generalization of Serre's conjecture to totally real fields has been formulated, and some relevant results have been obtained by Gee, Schein and others on the weights part of the conjecture. The level-lowering results of Ribet had been previously generalized to the setting of Hilbert modular forms in the work of Jarvis, Fujiwara and Rajaei.

In the advanced courses on Modularity several of these techniques and developments will be explained.

Goals: for goals of the course please check the following link

Scientific Committee

Henri Darmon (McGill University, Montreal)

Fred Diamond (King's College of London)

Luis Dieulefait (Universitat de Barcelona)

Bas Edixhoven (Leiden University)

Victor Rotger (Universitat Politècnica de Catalunya)

List of Speakers

Serre's conjectures and generalizations
Main Lecturer: Fred Diamond. Supplementary lectures by Luis Dieulefait.


p-adic Galois representations and global Galois deformations
Main Lecturers: Laurent Berger and Gebhard Böckle


Modularity Lifting Theorems and Potential Modularity
Main Lecturer: Jean-Pierre Wintenberger

Registration

Registration fee: 300€

Registration includes: attendance to the lectures, documentation package, copy of the course notes (in case of advanced courses), social dinner, cultural activity, and coffee breaks

Deadline for registration and payment:     May 14, 2010

Available Grants

Young researchers may apply for a grant and take advantge of a reduced registration fee and a lodging grant.
Awards are determined based on academic criteria and/or country of residence (giving special attention to advanced researchers from less favored countries).
There are two grants available:

Reduced registration fee for either type of grant holders: 90€

You will be informed as soon as possible whether support is available.

Deadline for grant applications:   April 20, 2010

Accommodation

Participants are encouraged to book their lodging as soon as possible.