Appears in collection : Summer School on the Langlands Program

We describe the construction of Galois representations associated to regular algebraic cuspidal automorphic representations of GL_n over a CM field, as well as those Galois representations associated to torsion classes that occur in the Betti cohomology of the corresponding locally symmetric spaces. The emphasis will be on Scholze’s proof, which applies to torsion classes and which uses perfectoid Shimura varieties and the Hodge-Tate period morphism.