Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type

Abstract

Abstract. For a Shimura variety of Hodge type with hyperspecial level structure at a prime p, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We de ne and study the Ekedahl-Oort strati cations on the special bers of those integral canonical models when p > . is generalizes Ekedahl-Oort strati cations de ned and studied by Oort on moduli spaces of principally polarized abelian varieties and those de ned and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements w in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to w is smooth of dimension l(w) (i.e. the length of w) if it is non-empty. We also determine the closure of each stratum.

Automorphic vector bundles with global sections on -schemes

“…3.4.4. Third, as C (1342) ∩ C (1243) ⊂ C (13) (24) , the equality of cones also holds for C (13) (24) , again by Prop. 3.4.4.…”

“…Let X be the special fiber of a Hodge-type Shimura variety with hyperspecial level at p. Let G be the corresponding reductive F pgroup and Z the zip datum of X. By [24], there is a smooth morphism of stacks ζ : X → G-Zip Z . Let X tor be a smooth, projective toroidal compactification of X afforded [15].…”

Automorphic Forms on the Stack of G-Zips

“…By abuse of notation, denote also by G the special fiber of a reductive Z p -model of G Qp and write S K := S K ⊗ κ(v) and F K := F K ⊗ κ(v). There is a smooth surjective map ζ : S K → G-Zip µ defined in [22,Th. 2.4.1], where µ is the cocharacter attached to the Shimura datum.…”

Quasi-constant characters: Motivation, classification and applications

“…Regarding the first problem, Shimura varieties of Hodge type furnish important examples of morphisms X → G-Zip µ . More precisely, suppose (G, X) is a Shimura datum of Hodge type, p is a prime at which G is unramified and (G, µ) arises from (G, [µ]) by reduction mod p. If K ⊂ G(A f ) is hyperspecial at p, then a theorem of Zhang asserts that there is a smooth morphism from the special fiber of the Kisin-Vasiu p-integral model of Sh(G, X) K to G-Zip µ , [27].…”