Ekedahl–Oort and Newton strata for Shimura varieties of PEL type
By E. Viehmann, T. Wedhorn. 2010
100 citations recorded by the source corpus.
We study the Ekedahl–Oort stratification for good reductions of Shimura varieties of PEL type. These generalize the Ekedahl–Oort strata defined and studied by Oort for the moduli space of principally polarized abelian varieties (the “Siegel case”). They are parameterized by certain elements $$w$$ in the Weyl group of the reductive group of the Shimura datum. We show that for every such $$w$$ the corresponding Ekedahl–Oort stratum is smooth, quasi-affine, and of dimension $$\ell (w)$$ (and in particular non-empty). Some of these results have previously been obtained by Moonen, Vasiu, and the second author using different methods. We determine the closure relations of the strata. We give a group-theoretical definition of minimal Ekedahl–Oort strata generalizing Oort’s definition in the Siegel case and study the question whether each Newton stratum contains a minimal Ekedahl–Oort stratum. As an interesting application we determine which Newton strata are non-empty. This criterion proves conjectures by Fargues and by Rapoport generalizing a conjecture by Manin for the Siegel case. We give a necessary criterion when a given Ekedahl–Oort stratum and a given Newton stratum meet.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/86e9808da02ed489ad93221d859132f5837b44d5
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Truncations of level 1 of elements in the loop group of a reductive group (2009, background): “roup over kF and a split maximal torus over a finite extension of kF can be lifted to a Borel subgroup and a split maximal torus over OF resp. over the corresponding unramified extension of OF, compare [VW], A.4. 4 2.2. The extended affine Weyl group fW has a decomposition Wf ∼= Ω ⋉W aff. Here Ω is the s”
- GEOMETRIC AND HOMOLOGICAL PROPERTIES OF AFFINE DELIGNE-LUSZTIG VARIETIES (2012, methodology): “There is a close relation between the affine Deligne-Lusztig variety X ˜ w ( b ) and the intersection of the Newton stratum associated with [ b ] with the Kottwitz-Rapoport stratum associated with ˜ w (see [10] and [32]).”
- Strata Hasse invariants, Hecke algebras and Galois representations (2015, background): “ratum in Sh1 K. The stratum Seis zero-dimensional. Zhang shows that the morphism ζis smooth (Theorem 3.1.2 in [Zha]), generalizing part of a result of Viehmann-Wedhorn in the PEL-case (Theorem 5.1 in [VW13]. For example, one then deduces the dimensions of the strata. However, contrary to Viehmann-We”
- Algebraic zip data (2010, methodology): “This is applied in [VW] to the study of Ekedahl-Oort strata for Shimura varieties of PEL type.”
- On the p-adic cohomology of the Lubin-Tate tower (2015, background): “vidence, RI, 2002. [19] L. Fargues, A. Genestier, and V. Lafforgue. L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, volume 262 of Progress in Mathematics. Birkh¨auser Verlag, Basel, 2008. [20] O. Gabber and L. Ramero. Almost ring theory, volume 1800 of Lecture Notes in Mathematics. Sprin”
- Stratifications in the reduction of Shimura varieties (2015, background): “This question is considered by Viehmann/Wedhorn [47] in the PEL-case, for hyperspecial level structures.”
- Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type (2013, methodology): “The next statement (including its proof) is a word by word adaptation of results in [28] (to be more precise, Proposition 4.”
- A relation between Dieudonne displays and crystalline Dieudonne theory (2010, methodology): “Relations between Dieudonné displays and crystalline Dieudonné theory 2203 This compatibility was not known before and can be useful in applications; see for example [Viehmann and Wedhorn 2013].”