# Modularity and potential modularity theorems in the function field setting

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Michael Harris, Columbia University
IAS Room S-101

Please note slight change in time (4:15).   Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of G.  The parameter is a homomorphism of the global Galois group into the Langlands L-group $^LG$ of G. I will report on my joint work in progress with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue's correspondence.  New (representation-theoretic and Galois-theoretic) issues arise when we seek to extend the earlier work of Böckle and Khare on the case of GL(n) to general reductive groups.  I describe hypotheses on the Langlands parameter that allow us to apply modularity arguments unconditionally, and I will state a potential modularity theorem for a general split adjoint group.