One of the most fundamental objects of arithmetic is the absolute Galois group of a number field. A rich source of representations for such groups is the p-adic cohomology of algebraic varieties. The Fontaine-Mazur conjecture predicts precisely which p-adic Galois representations ought to arise in this way. What is remarkable about the conjecture is that the most subtle condition in its formulation involves only the restriction of the Galois representation to the decomposition groups of primes above p. In certain situations Fontaine-Mazur combine their philosophy with that of Langlands, and predict which Galois representations come from modular eigenforms. Recently work on this conjecture has been given new impetus by a connection with Breuil's p-adic Langlands program, especially for representations which arise from higher weight modular forms. The project's aim is to pursue this connection by exploring new cases of this correspondence as well as their connections to geometry and applications to modularity.
A little over ten years ago Wiles proved Fermat's Last Theorem. He did this by relating elliptic curves to modular forms. The latter are complex functions which admit an incredibly large number of symmetries. Wiles' breakthrough involved the use of the p-adic Galois representation attached to an elliptic curve. His result can be viewed as a special case of a more general philosophy due to Fontaine and Mazur, which predicts that a certain class of p-adic Galois representations always arise from modular forms. Recently Breuil has proposed a p-adic analogue of the Langlands correspondence. This is a completely new ingredient which seems to have very strong applications to questions about modularity. The project aims to develop Breuil's correspondence and explore its relation to geometry and modularity of Galois representations.
