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 $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $\mathbf {Z}/p\mathbf {Z}$ -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 $\mathbf {Z}/p\mathbf {Z}$ -extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh.

Keywords

• 11S37
• 11F80