On the p-adic uniformization of unitary Shimura curves

Kudla, Stephen S.; Rapoport, Michael; Zink, Thomas

doi:10.48550/arxiv.2007.05211

Cited by 1 publication

(1 citation statement)

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“…In this way, they are able in certain cases to identify two RZ formal schemes for different (P)EL data which define closely related grouptheoretical data. For instance, using this approach they prove the conjectures of Rapoport-Zink [RZ17] and of Kudla-Rapoport-Zink [KRZ20] which postulated such hidden identifications, cf. [SW20, §25.4-25.5].…”

Section: Introductionmentioning

confidence: 99%

On integral local Shimura varieties

Pappas¹,

Rapoport²

2022

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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...

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Section: Introductionmentioning

confidence: 99%