Automorphic Galois representations and the cohomology of Shimura varieties

The first part of this report describes the class of representations of Galois groups of number fields that have been attached to automorphic representations. The construction is based on the program for analyzing cohomology of Shimura varieties developed by Langlands and Kottwitz. Using p-adic methods, the class of Galois representations obtainable in this way can be expanded slightly; the link to cohomology remains indispensable at present. It is often possible to characterize the set of Galois representations that can be attached to automorphic forms, using the modularity lifting methods initiated by Wiles a bit over 20 years ago. The report mentions some applications of results of this kind. The second part of the report explains some recent results on critical values of automorphic L-functions, emphasizing their relation to the motives whose `-adic realizations were discussed in the first part. Mathematics Subject Classification (2010). Primary 11F80; Secondary 11F70,