Geometrization of Local Langlands and the Cohomology of Shimura Varieties

Abstract

Fargues and Scholze showed that a candidate for the local Langlands correspondence forp-adic reductive groups G/Qp could be realized in terms of a geometric Langlands correspondence occurring over BunG, the moduli stack of G-bundles on the Fargues-Fontaine curve. This builds a bridge between Shimura varieties and the trace formula and shtukas and geometric Langlands, since the relevant shtuka spaces can be shown to uniformize Shimura varieties. The goal of this thesis is to build on this connection with the aim of describing the cohomology of Shimura varieties. The basic strategy is as follows. First, show that the Fargues-Scholze local Langlands correspondence agrees with more classical instances of the correspondence, by describing the Galois action on global Shimura varieties and using uniformization to relate this to shtukas. Second, combine such compatibility results with techniques in geometric Langlands to explicitly describe eigensheaves on BunG. Third, use this description to describe the cohomology of shtukas, and then use uniformization in the other direction to describe the cohomology of the global Shimura variety. In chapter 1, we showcase a strategy for showing that the Fargues-Scholze correspondence agrees with more classical instances of local Langlands in the particular case of GSp4and its inner form. We use this compatibility to describe eigensheaves with eigenvalue φ, a supercuspidal L-parameter, and in turn prove new cases of the Kottwitz conjecture. In chapter 2, we build on this paradigm, and show how, assuming such compatibility results, we can parabolically induce the eigensheaves on BunT for a maximal torus T ⊂ G to eigensheaves on BunG with eigenvalue factoring through a maximal torus. We do this under a generic assumption on the parameter, as in the work of Caraiani-Scholze. In chapter 3, we discuss joint work in progress where one combines the results of chapters 1 and 2 to extend the torsion vanishing results of Cariani-Scholze to several new cases. Motivated by this, we formulate several new conjectures on the cohomology of global Shimura varieties, and explain how these conjectures would follow from generalizing the analysis in chapter 2 to describe the eigensheaves with eigenvalue φ induced from a supercuspidal L-parameter factoring through the dual group of a general Levi subgroup of G