Local Shimura Varieties: Minicourse Given by Peter Scholze

This chapter discusses Peter Scholze's minicourse on local Shimura varieties. The goal of these lectures is to describe a program to construct local Langlands correspondence. The construction is based on cohomology of so-called local Shimura varieties and generalizations thereof. It was predicted by Robert Kottwitz that for each local Shimura datum, there exists a so-called local Shimura variety, which is a pro-object in the category of rigid analytic spaces. Thus, local Shimura varieties are determined by a purely group-theoretic datum without any underlying deformation problem. This is now an unpublished theorem, by the work of Fargues, Kedlaya–Liu, and Caraiani–Scholze. The chapter then explains the approach to local Langlands correspondence via cohomology of Lubin–Tate spaces as well as Rapoport–Zink spaces. It also introduces a formal deformation problem and describes properties of the corresponding universal deformation formal scheme.