Seminarios Ta de Números

Title: Geometric Hecke operators and moduli spaces of local shtukas

SPEAKER: Richard Magner (Boston University)

DATE & TIME: Tuesday, June 08th, 2021 - 17:30

ABSTRACT: The "fundamental curve of p-adic Hodge theory" introduced by Fargues-Fontaine some years ago provides a candidate for a nice geometric object bearing a relationship with the p-adic numbers analogous to the relationship between a smooth projective curve over a finite field and its global function field. In the latter setting, this relationship has been exploited to provide constructions to prove Langlands correspondences via techniques of "geometrization." Fargues conjectured that similar constructions should be possible in the p-adic setting, and recent work of Fargues-Scholze has provided significant progress, having now defined all constructions appearing in Fargues's conjectures carefully. In this talk, we will give an overview of these developments while providing some motivation coming from the cohomology theory of Rapoport-Zink spaces. We will also discuss how Fargues's conjectures relate to generalized conjectures on the cohomology of Rapoport-Zink space (or local shtuka spaces).