On integral 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 $(G,b,\hat{I}¼)$ 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 $(G,b,\hat{I}¼)$ is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any $(G,\hat{I}¼)$ of abelian type when $p\ne 2$, 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 $(G,b,\hat{I}¼,\mathcal{G})$.

61 pages, final version