The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation ρ : Galℚ → GL2(ℚ̅p) 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 ρ : Galℚ → GL2(L) when L is instead a non-archimedean local field of characteristic p.
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