ON INTEGRAL LOCAL SHIMURA VARIETIES

Abstract We give a construction of integral local Shimura varieties which are formal schemes that generalise 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 generalise the formal schemes defined by Rapoport-Zink via moduli of p -divisible groups, and are characterised 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 parametrises 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 fibre of this formal scheme to the local Shimura variety, a rigid-analytic space attached by Scholze to $(G, b, \mu , {\mathcal {G}})$ .