2020/09/29 by Tony Feng, Feng, Tony · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2009.14236
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 ℤ/pℤ-extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For ℤ/pℤ-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 ℤ/pℤ-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.