scholarly journals On the Global Structure of Special Cycles on Unitary Shimura Varieties

2013 ◽  
Vol 65 (5) ◽  
pp. 1125-1163
Author(s):  
Nicolas Vandenbergen
Keyword(s):  
Global Structure ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Local Models ◽  
Irreducible Components ◽  
Special Cycle

AbstractIn this paper, we study the reduced loci of special cycles on local models of the Shimura variety for GU(1; n − 1). Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat–Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.

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 On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

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.

The GL4 Rapoport–Zink Space

2020 ◽  
Author(s):  
Maria Fox
Keyword(s):  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Connected Components ◽  
Shimura Variety ◽  
Irreducible Components

Abstract We give a description of the $\textrm{GL}_4$ Rapoport–Zink space, including the connected components, irreducible components, intersection behavior of the irreducible components, and Ekedahl–Oort stratification. As an application of this, we also give a description of the supersingular locus of the Shimura variety for the group $\textrm{GU}(2,2)$ over a prime split in the relevant imaginary quadratic field.

scholarly journals Topological flatness of local models for ramified unitary groups. II. The even dimensional case

2013 ◽  
Vol 13 (2) ◽  
pp. 303-393 ◽  
Author(s):  
Brian D. Smithling
Keyword(s):  
Local Structure ◽  
Unitary Groups ◽  
Dimensional Case ◽  
Shimura Varieties ◽  
Local Models ◽  
Admissible Set ◽  
The Way ◽  
Moduli Problems

AbstractLocal models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure of integral models of certain$p$-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at${ \mathbb{Q} }_{p} $is ramified, quasi-split$G{U}_{n} $, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper, we proved that their new local models are topologically flat when$n$is odd. In the present paper, we prove topological flatness when$n$is even. Along the way, we characterize the$\mu $-admissible set for certain cocharacters$\mu $in types$B$and$D$, and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.

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 The Supersingular Locus of the Shimura Variety for GU(1, s)

2010 ◽  
Vol 62 (3) ◽  
pp. 668-720 ◽  
Author(s):  
Inken Vollaard
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Unitary Group ◽  
Shimura Variety ◽  
Incidence Relation ◽  
Irreducible Components ◽  
Reduction Modulo ◽  
Tits Building ◽  
Divisible Groups ◽  
Dieudonne Theory

AbstractIn this paper we study the supersingular locus of the reduction modulopof the Shimura variety for GU(1,s) in the case of an inert primep. Using Dieudonné theory we define a stratification of the corresponding moduli space ofp-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.In the case of GU(1, 2), we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

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.

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