Joint PU/IAS Number Theory

The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation $\rho:Gal_Q\rightarrow GL_2(\overline{Q}_p)$ is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations $\rho:Gal_Q\rightarrow GL_2(L)$ when L is instead a non-archimedean local field of characteristic p.