Regular de Rham Galois representations in the completed cohomology of modular curves
Let p be a prime. I want to explain how to use the geometry of modular curves at infinite level and the Hodge–Tate period map to study regular de Rham p-adic Galois representations appearing in the p-adically completed cohomology of modular curves. We will show that these Galois representations up to twists come from modular forms and give a geometric description of the locally analytic representations of GL2(Qp) associated to them. These results were previously known by totally different methods.