Local-global compatibility at l = p over CM fields (Oberseminar Arithm. Geometry and Representation Theory)

In joint work in progress with Hevesi, Thorne and Whitmore we prove that the global Galois representations associated with cuspidal, cohomological automorphic representations are compatible with the (semisimplified) local Langlands correspondence. In this talk I will explain the context and meaning of this statement. Moreover, I will sketch the proof of the main new technical ingredient, which is a uniform bound on the torsion in the cohomology of certain unitary Shimura varieties.