Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

Galois Representations

June 15 - July 10, 2009

University of Hawaii at Manoa, Honolulu, Hawaii

Organizers: Brian Conrad (Stanford), David Ellwood (CMI), Mark Kisin (Chicago), Chris Skinner (Princeton)

Many advances on the algebraic side of number theory in the last 15 years (such as the solutions of the Shimura-Taniyama conjecture, Sato-Tate conjecture and Serre's conjecture, as well as decisive progress on the Fontain-Mazur conjecture and Main Conjectures for modular forms) have relied in an essential way on improvements in the theory of Galois representations. For example, such improvements have enabled the local and global aspects of modularity lifting theorems to be extended far beyond the traditional 2-dimensional case over the rational numbers, and have led to generalizations of the "classical" theory of p-adic modular forms in a way that makes more effective use of representation theory and geometry to obtain results on the arithmetic of L-values.

The aim of the three main courses is to present an overview of many of these ideas and applications, aimed at advanced graduate students and post docs with a strong background in number theory, Galois cohomology, and basic algebraic geometry.  One course will focus entirely on local problems (p-adic representations of Galois groups of p-adic fields), a second course will  have a more global flavor (Galois deformation theory and global applications), and a third (on L-values) will rely on the other two courses.  In the final week of the program there will be three mini-courses that build on themes introduced in the foundational courses.  These will address aspects of the following topics: proofs of p-adic comparison isomorphisms (Andreatta), introduction to the p-adic Langlands correspondence (Emerton), and construction of global p-adic Galois representations Shin).

Foundational Courses

p-adic Hodge Theory
Oliver Brinon (Paris 13), Brian Conrad (Stanford)
(Φ, Γ)-modules, applications to potentially semi-stable deformation rings and families of Galois representations

Deformation of Galois Representations and Modular Forms
Mark Kisin (Chicago), Jacques Tilouine (Paris 13)
Deformation theory of Galois representations, pseudo-representations, with applications to eigenvarieties, construction of Galois respresentations, Taylor-Wiles method

Iwasawa Theory and Automorphic Applications
Joel Bellaiche (Brandeis), Chris Skinner (Princeton)
Iwasawa theory and Hida theory with applications to Selmer groups, automorphic forms, and the arithmetic of special values of L-functions

Mini Courses

Proofs of p-adic Comparison Isomorphisms
Fabrizio Andreatta (Milan)

Introduction to the p-adic Langlands Correspondence
Matthew Emerton (Northwestern)

Construction of global p-adic Galois Representations
Sug Woo Shin (Chicago)