Smith theory and Langlands functoriality

According to Langlands’ conjectures, algebraic automorphic forms should be parametrized by global Galois representations, and smooth representations of pp-adic groups should be parametrized by local Galois representations. These parametrizations have recently been constructed in the function field context by Vincent Lafforgue in the global setting, and Genestier-Lafforgue in the local setting. Using recent tools in modular representation theory, including Smith theory and parity sheaves, we prove that these parametrizations enjoy certain expected functoriality properties. Globally, we establish the existence of functorial transfers of mod p automorphic forms through pp-cyclic base change. Locally, we prove that Tate cohomology realizes cyclic base change functoriality in the mod pp Genestier-Lafforgue correspondence, verifying a function field version of a conjecture of Treumann-Venkatesh.