Geodesic
Smith theory and cyclic base change functoriality
Forum of Mathematics, Pi, Tome 12 (2024)

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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.The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular 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. 
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Tony Feng. Smith theory and cyclic base change functoriality. Forum of Mathematics, Pi, Tome 12 (2024). doi: 10.1017/fmp.2023.32

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