Modularity and Reciprocity: a Robust Approach

Introduction

Many of the most important questions in number theory and arithmetic geometry can be approached through the theory of Galois representations. A powerful tool to understand such representations is the Langlands programme, which describes the conjectural relations between Galois representations and automorphic forms. Landmark results in this direction include the proof of the modularity conjecture for (the Galois representations associated to) elliptic curves over the field of rational numbers and the proof of Serre's conjecture.

Advances in Understanding

Dramatic advances in our understanding of the structures of the Langlands programme in the last 20 years have made it possible to extend the scope of these theorems, both to more general classes of Galois representations and to more general base number fields. However, the most general and conclusive statements remain out of reach, in large part due to our poor understanding of Galois deformation theory in the most degenerate situations.

Proposal Goals

The goal of this proposal will be to address central questions in the arithmetic of the Langlands programme by introducing new techniques into the study of Galois representations that are robust, powerful, and flexible.

Approach

We will take a multi-faceted and cohesive approach that will lead to:

1. A greater understanding of fundamental open questions.
2. The proofs of new cases of important conjectures in arithmetic geometry and the theory of automorphic forms, including:
   • The Fontaine--Mazur conjecture
   • The general form of Serre's conjecture
   • The Langlands functoriality conjectures.