scholarly journals Vanishing theorems for the mod p cohomology of some simple Shimura varieties

2020 ◽  
Vol 8 ◽  
Author(s):  
Teruhisa Koshikawa
Keyword(s):  
Additional Assumption ◽  
Hecke Algebra ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Vanishing Theorems ◽  
Low Dimensions ◽  
Eisenstein Ideal ◽  
Mod P

Abstract We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

scholarly journals SUR LA TORSION DANS LA COHOMOLOGIE DES VARIÉTÉS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

2017 ◽  
Vol 18 (3) ◽  
pp. 499-517 ◽  
Author(s):  
Pascal Boyer
Keyword(s):  
Maximal Ideal ◽  
Cohomology Class ◽  
Hecke Algebra ◽  
Galois Representation ◽  
Local System ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Torsion Freeness ◽  
Satake Parameters ◽  
Torsion Free

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

scholarly journals On Goren–Oort stratification for quaternionic Shimura varieties

2016 ◽  
Vol 152 (10) ◽  
pp. 2134-2220 ◽  
Author(s):  
Yichao Tian ◽  
Liang Xiao
Keyword(s):  
Line Bundle ◽  
Real Field ◽  
Necessary Condition ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Maximal Level ◽  
Totally Real ◽  
Totally Real Field

Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic-$p$ fiber of a quaternionic Shimura variety of maximal level at $p$. We show that each stratum is a $(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer $r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$.

scholarly journals Local models for Weil-restricted groups

2016 ◽  
Vol 152 (12) ◽  
pp. 2563-2601 ◽  
Author(s):  
Brandon Levin
Keyword(s):  
Hecke Algebra ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Local Models ◽  
Central Function ◽  
Nearby Cycles ◽  
Weil Restriction ◽  
Trace Of Frobenius ◽  
Invent Math

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

scholarly journals GALOIS CONJUGATES OF SPECIAL POINTS AND SPECIAL SUBVARIETIES IN SHIMURA VARIETIES

2019 ◽  
pp. 1-15
Author(s):  
Martin Orr
Keyword(s):  
Special Point ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Canonical Models ◽  
Special Points

Let $S$ be a Shimura variety with reflex field $E$ . We prove that the action of $\text{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

scholarly journals Local models in the ramified case. III Unitary groups

2009 ◽  
Vol 8 (3) ◽  
pp. 507-564 ◽  
Author(s):  
G. Pappas ◽  
M. Rapoport
Keyword(s):  
Local Structure ◽  
Level Structure ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Flag Varieties ◽  
Shimura Variety ◽  
Local Models ◽  
Affine Flag

AbstractWe continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primespat which the group defining the Shimura variety ramifies. We describe ‘good’p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

scholarly journals The μ-ordinary Hasse invariant of\break unitary Shimura varieties

2017 ◽  
Vol 2017 (728) ◽  
Author(s):  
Wushi Goldring ◽  
Marc-Hubert Nicole
Keyword(s):  
Shimura Varieties ◽  
Shimura Variety

AbstractWe construct a generalization of the Hasse invariant for any Shimura variety of PEL-type

scholarly journals On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

2022 ◽  
Author(s):  
Fabrizio Andreatta
Keyword(s):  
Flag Variety ◽  
Integral Model ◽  
Shimura Variety ◽  
Period Map ◽  
Mod P

AbstractConsider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime $$p\ge 5$$ p ≥ 5 . Consider its perfectoid cover $$S^{\text {ad}}(p^\infty )$$ S ad ( p ∞ ) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to $$S^{\text {ad}}(p^\infty )$$ S ad ( p ∞ ) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to $$S^\text {ad}(p^\infty )$$ S ad ( p ∞ ) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.

scholarly journals Foliations on unitary Shimura varieties in positive characteristic

2018 ◽  
Vol 154 (11) ◽  
pp. 2267-2304 ◽  
Author(s):  
Ehud de Shalit ◽  
Eyal Z. Goren
Keyword(s):  
Tangent Bundle ◽  
Positive Characteristic ◽  
Blow Up ◽  
Level Structure ◽  
Horizontal Component ◽  
Zariski Closure ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Integral Manifolds ◽  
Imaginary Field

When$p$is inert in the quadratic imaginary field$E$and$m<n$, unitary Shimura varieties of signature$(n,m)$and a hyperspecial level subgroup at$p$, carry a naturalfoliationof height 1 and rank$m^{2}$in the tangent bundle of their special fiber$S$. We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem$S^{\sharp }$, a successive blow-up of$S$. Over the ($\unicode[STIX]{x1D707}$-)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of$S^{\sharp }$by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber$S_{0}(p)$of a certain Shimura variety with parahoric level structure at$p$. As a result, we get that this ‘horizontal component’ of$S_{0}(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature$(m,m)$, and a certain Ekedahl–Oort stratum that we denote$S_{\text{fol}}$. We conjecture that these are the only integral submanifolds.

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

2018 ◽  
Vol 70 (2) ◽  
pp. 451-480 ◽  
Author(s):  
Chao Zhang
Keyword(s):  
Weyl Group ◽  
Moduli Spaces ◽  
Level Structure ◽  
Reductive Group ◽  
Extension Property ◽  
Integral Model ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Canonical Models ◽  
Special Fibers

AbstractFor 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 define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when p > 2. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined 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.

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