Local-global compatibility over function fields

Abstract: The Langlands program predicts a relationship between automorphic representations of a reductive group $G$ and Galois representations valued in its $L$-group. For general $G$ over a global function field, the automorphic-to-Galois direction has been constructed by V. Lafforgue. More recently, for general $G$ over a nonarchimedean local field, a similar correspondence has been constructed by Fargues–Scholze.

We present a proof that the V. Lafforgue and Fargues–Scholze correspondences are compatible, generalizing local-global compatibility from class field theory. As a consequence, the correspondences of Genestier–Lafforgue and Fargues–Scholze agree, which answers a question of Fargues–Scholze, Hansen, Harris, and Kaletha.

number theory

Audience: researchers in the topic

Harvard number theory seminar