Modularity and potential modularity theorems in the function field setting

Modularity and potential modularity theorems in the function field setting - Harris

Michael Harris
Columbia University
December 17, 2015
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 $\mathrm{GL}(n)$ 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.