Appears in collection : Reductive groups and automorphic forms. Dedicated to the French school of automorphic forms and in memory of Roger Godement.

Let K be an imaginary quadratic field. Conjecturally, one should be able to associate to any cusp form on GL_n(A_K) which is cohomological (for the trivial coefficient system) a Galois representation. This can be achieved using our understanding of the classification of automorphic representations of the quasi- split unitary group U(n, n), which relies upon the stabilization of the twisted trace formula for GL_n. A detailed understanding of the local properties of these Galois representations opens up the possibility of proving automorphy lifting theorems. I will describe work in progress of a 10 author collaboration that proves such theorems, using as a starting point very important vanishing theorems for the cohomology of non-compact Shimura varieties which are work in progress of Caraiani--Scholze. A particular consequence is the Ramanujan conjecture for cohomological cusp forms in the case n = 2. (The 10 authors are Allen, F. Calegari, Caraiani, Gee, Helm, Le Hung, J. Newton, Scholze, Taylor, and myself.)