The Fontaine-Mazur conjecture via p-adic modular forms.
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. A few years ago Coleman and Mazur, building on work of Hida, discovered that modular eigenforms tend to move in p-adic families. The aim of the project is to approach the Fontaine-Mazur conjecture using the geometry of these families, and corresponding families of Galois representations: The modularity of some Galois representations can be proved following ideas of Wiles and Taylor-Wiles, and then one hopes to deduce the modularity of the rest by p-adic interpolation style arguments.
Almost 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. My aim is to approach this conjecture by exploiting the fact discovered by Hida and Coleman-Mazur - that modular forms tend to move in p-adic families. This is rather surprising given that they are complex valued functions. It serves as a striking illustration that although modular forms are defined as analytic objects, they are of a profoundly arithmetic nature.
