Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of Q_p. Rapoport and Zink have defined a model of the Shimura variety over the ring of integers of the completion of the reflex field at a place lying over p, with parahoric level structures at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced.
Level $m$-stratifications on PEL Shimura varieties are defined and studied by Wedhorn using BT-$m$s with PEL structure, and then by Vasiu for general Hodge type Shimura varieties using Shimura $F$-crystals. The theory of foliations is established by Oort for Siegel modular varieties, and by Mantovan for PEL Shimura varieties. It plays an important role in Hamacher's work to compute the dimension of Newton strata of PEL Shimura varieties. We study level $m$ stratifications on good reductions at $p>2$ of Shimura varieties of Hodge type by constructing certain torsors together with equivariant mo…
We give a simple proof that Kottwitz's PEL type integral models of Shimura varieties admit closed embeddings into Siegel integral models. We also show that Rapoport's and Kottwitz's integral models agree with Kisin's integral models for relevant Shimura data.
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oort's method for Siegel modular varieties.
We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.
We study integral models, so-called Pappas-Rapoport or splitting models, of some PEL Shimura Varieties whose data are ramified at a prime p. We show that except in a specific case, these models are smooth when there is no level at p, and we study their special fiber modulo p. We show that interesting stratifications appear, and we show that the $μ$-ordinary locus is open and dense, following a strategy of Wedhorn in the unramified case.
The aim of this article is the construction of the interior motive of a Picard variety. Those are Shimura varieties of PEL type. Our result is an application of the strategy developed by Wildeshaus to construct a Hecke-invariant motive whose realizations correspond to interior cohomology.
For infinitely many Hurwitz spaces parametrizing cyclic covers of the projective line, we provide a method to determine the integral PEL datum of the Shimura variety that contains the image of the Hurwitz space under the Torelli morphism.
In this paper we study the geometry of good reductions of Shimura varieties of abelian type. More precisely, we construct the Newton stratification, Ekedahl-Oort stratification, and central leaves on the special fiber of a Shimura variety of abelian type at a good prime. We establish several basic properties of these stratifications, including the non-emptiness, closure relation and dimension formula, generalizing those previously known in the PEL and Hodge type cases. We also study the relations between these stratifications, both in general and in some special cases, such as those of fully H…
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 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 give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a consequence (of our theorem 9.1) one obtains that for every prescribed odd prime characteristic $p$ every bounded symmetric domain possesses quotients by arithmetic groups whose models have good reduction at a prime divisor of $p$.
Topological flatness of local models for ramified unitary groups. I. The odd dimensional case
局部模型是根据线性代数模问题定义的某些方案，其给出由Rapoport和Zink定义的某些p-adic PEL Shimura簇的积分模型的étale局部邻域。当定义志村簇的群在p处分歧时，最初定义的局部模型（以及志村模型）可能不是平坦的，并且需要修改它们的定义以获得平坦的方案。在酉相似群在Q_p处的局部化是分歧的情况下，准分裂GU_n，Pappas和Rapoport在定义原局部模型的模问题上增加了新的条件，即所谓的楔条件和旋条件，并证明了新的局部模型是平坦的。我们证明了他们猜想的一个初步形式，即他们的新模型是拓扑平坦的，在n是奇数的情况下。
Congruence relations for Shimura varieties associated to some unitary groups
本文研究了与签名为（n-1，1）的酉群相关的PEL-Shimura簇在惰性和非分歧情形下的约化.我们描述了牛顿多边形和Ekedahl-Oort分层。我们进一步研究了同构的相关模空间，并使用局部模型的一个变体给出了一个通用的描述。作为应用，我们证明了Blasius und Rogawski关于整数n为偶数时的同余关系的一个猜想.
We define variants of PEL type of the Shimura varieties that appear in the context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.
We show that the zip stratification given by an arbitrary $\Ghat$-zip over a scheme is pure. We deduce purity of the level-$m$-stratification for truncated Barsotti-Tate groups and purity of the Ekedahl-Oort stratification for special fibers of good models of Shimura varieties of Hodge type. We also show that all Ekedahl-Oort strata are quasi-affine schemes and generalize several properties of the Bruhat stratification from the PEL case to the case of Shimura varieties of Hodge type.
This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori-Hecke algebras. This is commonly referred to as Kottwitz's Conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even di…
We construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywh…
We establish vanishing results for spaces of automorphic forms in characteristic $0$ and characteristic $p$. We prove that for Hodge-type Shimura varieties, the weight of any nonzero automorphic form in characteristic $0$ satisfies the Griffiths-Schmid conditions, by purely algebraic, characteristic $p$ methods. We state a conjecture for general Hodge-type Shimura varieties regarding the vanishing of the space of automorphic forms in characteristic $p$ in terms of the weight. We verify this conjecture for unitary PEL Shimura varieties of signature $(n-1,1)$ at a split prime.
In this article, we treat two questions on Rapoport--Zink spaces of Hodge type constructed by Hamacher and Kim. One of which is their singularities, and the other is $p$-adic uniformization of Shimura varieties. More precisely, we prove that the singularity of a Rapoport--Zink space is controlled by its asssociated local model, and the basic locus of a Kisin--Pappas integral model of a Shimura variety is uniformized by the corresponding Rapoport--Zink space. These results extend the known facts by Rapoport and Zink in the case of PEL type.
Topological flatness of local models for ramified unitary groups. II. The even dimensional case
局部模型是根据线性代数模问题定义的方案，它给出了由Rapoport和Zink定义的某些p-adic PEL Shimura簇的积分模型的étale局部邻域。对于在Q_p上的局部化是分歧的么正相似群（准分裂GU_n），帕帕斯观察到原始局部模型通常不是平坦的，并且他和拉波波特为原始模问题引入了新的条件，他们推测这会产生平坦的方案。在以前的文件中，我们证明了他们的新的局部模型是拓扑平坦的，当n是奇数。本文证明了n为偶数时的拓扑平坦性。一路上，我们的特征的mu-容许集的某些cocharacters亩在类型B和D，我们表明，这些cocharacters的容许性可以在顶点的方式，证实了一个猜想的帕帕斯和Rapoport。
We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s$-conjugacy classes. As an application, we verify the Axioms in \cite{HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s$-conjugacy class in a residually split group, the set of connected components is "controlled" by the set of straight elements associated to the $\s$-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with \…
