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Thursday, February 25, 2016

**Abstract:** The Langlands correspondence for number fields predicts a precise distinguished subclass of all automorphic representations that should be in correspondence with pure motives (and so also with their various realizations: certain Galois representations, pure Hodge structures, etc.). In particular, many automorphic representations which are initially defined by analytic and/or representation-theoretic means should have deep algebraic properties, the simplest being that their Hecke eigenvalues should be algebraic. In this talk, I will focus on the possibilities and limitations of using Langlands functoriality to prove such algebraicity results. I'll begin by explaining how automorphic representations naturally break-up into several classes, which are motivated by both geometry and representation theory. I will then discuss some general negative results, some positive examples and some open problems about when it is possible to ``move'' from one of these classes to another one by means of functoriality.