Abstract
When$p>2$, we construct a Hodge-type analogue of Rapoport–Zink spaces under the unramifiedness assumption, as formal schemes parametrizing ‘deformations’ (up to quasi-isogeny) of$p$-divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of ‘local Shimura varieties’ conjectured by Rapoport and Viehmann.
Highlights
In the case of unramified EL- and PEL-type, we show that RZG,b recovers the original construction of Rapoport–Zink space in [44, Theorem 3.25]
Let G be a reductive group over R, and Λ a finite free R-module equipped with a closed immersion of algebraic R-groups G → GL(Λ). (We identify G with its image in GL(Λ).) there exist finitely many elements sα ∈ Λ⊗ such that G coincides with the pointwise stabilizer of; that is, for any R-algebra R we have
We introduce a notion of filtrations on E which etale-locally admits a splitting given by some cocharacter; see Definition 4.6 for the relevant setting
Summary
Let (G, H) be a Hodge-type Shimura datum; that is, (G, H) can be embedded into the Shimura datum associated to some symplectic similitude group (that is, Siegel Shimura datum). We choose an element b ∈ G(Qupr) which gives rise to a p-divisible group X over Fp in the following sense: for some finite free Zp-module Λ with faithful G-action, the F-crystal M := (Zupr ⊗ Λ∗, b ◦ (σ ⊗ id)) gives rise to a p-divisible group X by the (contravariant) Dieudonnetheory In this case, we can associate to such b an ‘unramified Hodge-type local Shimura datum’ (G, [b], {μ−1}) (cf Section 2.5). There exists a closed formal subscheme RZG,b ⊂ RZX which classifies deformations (up to quasi-isogeny) of X with Tate tensors (tα), such that the Hodge filtration of the p-divisible group is etale-locally given by some cocharacter in the conjugacy class {μ}. Oliver Bultel and George Pappas [9] gave yet another purely grouptheoretic (and local) construction of Hodge-type Rapoport–Zink spaces via the theory of (G, μ)-displays if the fixed p-divisible group X does not have any nontrivial multiplicative or etale part.
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