This project is concerned with the correspondence, predicted by Langlands and Fontaine-Mazur, between Galois representations and automorphic representations. This correspondence is still largely conjectural but there has been much progress in recent years. The proof of the Shimura-Taniyama conjecture has been considerably generalized to rather general potential automorphy theorems that apply to Galois representations of arbitrary dimension. Such results have led, for example, to a proof of the Sato-Tate conjecture. On the other hand, these results have the restriction that they apply only to Galois representations with distinct Hodge-Tate weights. This restriction rules out some well known examples of Galois representations such as Artin representations and the Tate modules of abelian varieties of dimension greater than one. The principal goal of this project is to develop modularity results that apply to Galois representations with repeated Hodge-Tate weights.
Number theory is one of the oldest branches of mathematics and has a very rich history. Modern number theorists are interested in the field of algebraic numbers (numbers that satisfy a polynomial equation with rational coefficients) and its group of symmetries (which is an example of a Galois group). The conjectures of Langlands and Fontaine-Mazur then predict that representations of this group are intimately related to objects from distinct branches of mathematics such as representation theory, analysis and algebraic geometry. Progress on these conjectures has led to some spectacular results such as Wiles' proof of Fermat's Last Theorem and has been a driving force for much development in neighboring fields. Objects of fundamental interest in this theory, such as elliptic curves, play a large role in cryptography and internet security.
