Equivariant localization, parity sheaves, and cyclic base change

Abstract: Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. I will explain some recently established properties of these correspondences regarding base change functoriality: existence of transfers for mod $p$ automorphic forms through $p$-cyclic base change in the global correspondence, and Tate cohomology realizes $p$-cyclic base change in the mod $p$ local correspondence. In particular, the local statement verifies a conjecture of Treumann-Venkatesh. The proofs combine Lafforgue’s theory with equivariant localization arguments for shtukas as well as recent advances in modular representation theory, namely parity sheaves and Smith-Treumann theory. Compared with previous iterations of the talk, this time the talk will emphasize the role of the new representation-theoretic tools, during the extra 20 minutes.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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