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 pdivisible groups, and are characterized purely in group-theoretic terms.More precisely, for a local p-adic Shimura datum (G, b, µ) and a quasi-parahoric group scheme G for G, Scholze has defined a functor on perfectoid spaces which parametrizes padic shtukas. He conjectured that this functor is representable by a normal formal scheme which is locally formally of finite type and flat over O Ȇ . Scholze-Weinstein proved this conjecture when (G, b, µ) is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any (G, µ) of abelian type when p = 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, µ, G).In fact, we prove that the decomposition (1.0.2) comes by passing to the generic fiber of a decomposition of functorsHere G o β denotes the parahoric group scheme associated to the quasi-parahoric group scheme G β (the neutral connected component). This formula also gives a reduction of Scholze's conjecture from quasi-parahoric group schemes to parahoric group schemes.We know quite a bit about the local structure of the formal scheme M G,b,µ . Indeed, letµ be the local model associated to the parahoric group scheme G o associated to G. Here the associated v-sheaf on Perfd k is defined in [SW20, §21] and the representability by a weakly normal scheme M loc G o ,µ flat over O Ȇ is established in the case of a general local Shimura datum by J. Anschütz, I. Gleason, J. Lourenço, T. Richarz in [AGLR22]. In fact, M loc G o ,µ is always normal with reduced special fiber (by [AGLR22] and [GLo22] which settled some remaining cases for p = 2 and p = 3). When p = 2, this local model also coincides for local Shimura data of abelian type with the local model in the style of Pappas and Zhu [PZ13] (modified in [HPR20]), as extended to groups which arise by restriction of scalars from wild extensions by B. Levin [Le01]. When p = 2 and G ad is of type A or C, this local model also coincides with the local model obtained by taking the closure of the generic fiber in the naive local model of [RZ96]. We also know that, if p = 2, M loc G,µ is Cohen-Macaulay [HR19]. Then we prove under our assumptions above that for every x ∈ M G,b,µ (k), there exists y ∈ M loc G,µ (k) and an isomorphism of formal completions M G,b,µ /x ≃ M loc G,µ/y .When G is a parahoric group scheme, Gleason has defined the formal completion M int G,b,µ /x as a v-sheaf. In our approach, we first show that his definition extends to the case of an arbitrary quasi-parahoric group scheme and then show that M int G,b,µ /x is representable (by the formal spectrum of a complete Noetherian loca...