Joint IAS/Princeton University Number Theory Seminar

In this talk I will explain some recent results on local-global compatibility for the action of GL_2 of the finite adeles on the p-adically completed cohomology of modular curves, as well as some applications of these results (e.g. to the Fontaine-Mazur conjecture). These results rely for their proof on the existence of a p-adic local Langlands correspondence, recently constructed by Colmez, and I will also spend some time describing Colmez's work.