Smith theory and cyclic base change functoriality

Abstract

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbb{Z}/p\mathbb{Z}$-extensions of global function fields, we prove the existence of base change for mod $p$ automorphic forms on arbitrary reductive groups. For $\mathbb{Z}/p\mathbb{Z}$-extensions of local function fields, we construct a base change homomorphism for the mod $p$ Bernstein center of any reductive group. We then use this to prove existence of local base change for mod $p$ irreducible representation along $\mathbb{Z}/p\mathbb{Z}$-extensions for all large enough $p$, and that Tate cohomology realizes descent along base change, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod $p$ spherical Hecke algebras, in a joint appendix with Gus Lonergan.