On integral local Shimura varieties

TL;DR

This paper constructs integral local Shimura varieties as formal schemes generalizing known models, proves Scholze's conjecture for a broad class of cases, and relates these schemes to local Shimura varieties.

Contribution

It provides a new construction of integral local Shimura varieties for all classical groups at odd p and proves Scholze's conjecture in abelian type cases for p ≠ 2.

Findings

Construction of integral local Shimura varieties for classical groups.

Proof of Scholze's conjecture for abelian type cases when p ≠ 2.

Relation established between the formal schemes and local Shimura varieties.

Abstract

We give a construction of "integral local Shimura varieties" which are formal schemes that generalize the well-known integral models of the Drinfeld $p$-adic upper half spaces. The construction applies to all classical groups, at least for odd $p$. These formal schemes also generalize the formal schemes defined by Rapoport-Zink via moduli of $p$-divisible groups, and are characterized purely in group-theoretic terms. More precisely, for a local $p$-adic Shimura datum $(G, b, \mu)$ and a quasi-parahoric group scheme $\mathcal G$ for $G$, Scholze has defined a functor on perfectoid spaces which parametrizes $p$-adic shtukas. He conjectured that this functor is representable by a normal formal scheme which is locally formally of finite type and flat over $O_{\breve E}$. Scholze-Weinstein proved this conjecture when $(G, b, \mu)$ is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any $(G, \mu)$ of abelian type when $p\neq 2$, and when $p=2$ and $G$ is of type $A$ or $C$. We also relate the generic fiber of this formal scheme to the local Shimura variety, a rigid-analytic space attached by Scholze to $(G, b, \mu, {\mathcal G})$.