Local-global compatibility over function fields

Abstract

We prove that V. Lafforgue's global Langlands correspondence is compatible with Fargues-Scholze's semisimplified local Langlands correspondence. As a consequence, we canonically lift Fargues-Scholze's construction to a non-semisimplified local Langlands correspondence for positive characteristic local fields. We also deduce that Fargues-Scholze's construction agrees with that of Genestier-Lafforgue, answering a question of Fargues-Scholze, Hansen, Harris, and Kaletha. The proof relies on a uniformization morphism for moduli spaces of shtukas.

Removed restrictions on $p$ in Theorem B and fixed minor errors. 54 pages. Comments welcome!

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