Advanced Courses on Modularity

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.

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)

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

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