Adrian Vasiu
Let $(G,X)$ be a Shimura variety of PEL type such that $G_{{\bf Q}_2}$ is a split ${\bf GSO}_{2n}$ group with $n\ge 2$. We prove the existence of the integral canonical models of ${\rm Sh}(G,X)/H_2$ in unramified mixed characteristic $(0,2)$, where $H_2$ is a hyperspecial subgroup of $G({\bf Q}_2)$.
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We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
We construct relative PEL type embeddings in mixed characteristic (0,2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0,2) of hermitian orthogonal Shimura varieties of PEL type.
The aim of this paper is to generalize results of C.-L. Chai about the monodromy of Hecke invariant subvarieties to Shimura varieties of PEL-type.
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.
In this paper, we compare two cohomology groups associated to Shimura varieties of PEL type, which are not proper over the base. One is the compactly supported l-adic cohomology, and the other is the nearby cycle cohomology, namely, the compactly supported cohomology of the nearby cycle complex for the canonical integral model of the Shimura variety over Z_p. We prove that the G(Q_p)-cuspidal part of these cohomology groups are the same, where G denotes the reductive algebraic group naturally attached to the PEL datum. Some applications to unitary Shimura varieties are also given.
Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I
该研究证明了任意非分歧混合特征(0,p)下Hodge型志村簇良好光滑整模型的存在性，给出Langlands猜想的光滑解，并推进了Milne和Reimann相关猜想的证明，证明方法新颖且更具原理性。
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.
The Bruhat stratification for Shimura varieties of PEL type is studied. In the Siegel case this stratification is a scheme-theoretic variant of the stratification by the a-number. We show that all Bruhat strata are smooth and determine their dimensions. We also prove that the closure of a Bruhat stratum is a union of Bruhat strata and describe which Bruhat strata are contained in the closure of a given Bruhat stratum.
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.
Explicit Kodaira-Spencer maps over PEL Shimura varieties
本文通过Kodaira-Spencer映射构造两个canonical线丛间的映射，并显式计算其对线丛canonical度量的影响，为比较两个canonical线丛及其算术交数提供了具体方法。
We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$-rank stratification. We show that the $p$-rank is constant on a KR stratum, generalizing a result of Ngô and Genestier. We prove an abstract, uniform formula for the $p$-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$-rank 0 locus.
On the Kudla-Rapoport conjecture for unitary Shimura varieties with maximal parahoric level structure at unramified primes
本文通过 Deligne-Lusztig 簇的 Tate 猜想版本证明 Kudla-Rapoport 猜想，并在任意维数的若干新情形下无条件成立。
We study $p$-adic integral models of certain PEL Shimura varieties with level subgroup at $p$ related to the $Γ_1(p)$-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of $\mathbb{Q}_p$ and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are étale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort-Tate generator of a group scheme. We use these local models t…
We prove the Mumford--Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prove the conjecture for the orthogonal case (i.e., for the $B_n$ and $D_n^R$ Shimura types). As a main tool, we construct embeddings of Shimura varieties (whose adjoints are) of prescribed abelian type into unitary Shimura varieties of PEL type. These constructions implicitly classify the adjoints of Shimura varieties of PEL type.
We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of integral canonical models of Shimura varieties of preabelian (resp. of abelian) type in mixed characteristic $(0,p)$ with $p\Ge 3$ (resp. with $p=2$) and with respect to hyperspecial subgroups; if $p=3$ (resp. if $p=2$) we restrict in this part I either to the $A_n$, $C_n$, $D_n^{\dbH}$ (resp. $A_n$ and $C_n…
We study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms. This generalises the work of Coleman, and is a first step in an ongoing project to extend the cohomological approach to classicality to higher-dimensional Shimura varieties.
The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n-1) over Q is uniformized by a formal scheme \Cal N. In the case when p is inert, we define special cycles Z(x) in \Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional -- in which case it reduce…
The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$
本文通过使用Casselman-Shalika公式，提供了对于SL(2)×GL(2)的Shimura型Rankin-Selberg积分的无分歧计算的新证明。
Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative approach to vanishing results of Caraiani-Scholze for the cohomology of unitary Shimura varieties.
We prove the existence of integral canonical models of unitary Shimura varieties in arbitrary unramified mixed characteristic. Errata to [Va1] are also included.
We construct Eigenvarieties for PEL Shimura varieties which interpolate cuspidal, finite slope automorphic forms for PEL Shimura varieties appearing as global sections of (coherent) automorphic sheaves, under the hypothesis that the primes are unramified in the Shimura datum, but allowing the ordinary locus to be empty. We then use this construction to extend previous results on the Bloch-Kato conjecture for Hecke characters of a quadratic imaginary field.
A PEL-type Igusa Stack and the $p$-adic Geometry of Shimura Varieties
本文通过构造一个称为 Igusa 叠的 v-叠，将无限 p-级 Shimura 簇的良好约化区域描述为与仿射格拉斯曼流形的 Schubert 胞腔在 Fargues-Fontaine 曲线上的纤维积，并推广到极小紧化，同时当 Schubert 胞腔替换为有界子叠时恢复积分模型。
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type.
Let p>2 be a prime and let X be a compactified PEL Shimura variety of type (A) or (C) such that p is an unramified prime for the PEL datum. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic p-adic modular forms of Iwahoric level for X. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected…
Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(\mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).
Entire theta operators at unramified primes
本文针对PEL型Shimura簇上的自守形式theta算子，实现了Maass--Shimura算子模p约化在非分歧素数处的解析延拓，构造了新型模p theta算子，并推广至任意签名与任意次数CM域。
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 note, we study Shimura varieties for the groups $\mathrm{GU}(V)$, where $V$ is a Hermitian space relative to a CM extension $E/E^+$. We give a description of the supersingular locus of the fiber at a prime $ν$ over $p$ of such a Shimura variety, under the assumptions that $\mathrm{dim}_E(V) \leq 4$ and that the prime $p$ splits completely in $E^+$ and is unramified in $E$.
We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic cove…
Geometry around unramified CM points
本文通过构造特殊affinoid并应用正深度Deligne-Lusztig理论，计算了Yu构造的超尖表示对局部Shimura簇中间上同调的显式贡献，并利用局部shtukas叠解决了典型子群问题。
Igusa stacks for certain abelian-type Shimura varieties
本文构造了特定abel-type Shimura varieties的良好约简区域的Igusa堆栈，基于PEL数据并假设类型为(A even)或(C)且在素数p处无分支。
We prove that in the unramified case, local models of Shimura varieties with Iwahori level structure are normal and Cohen Macaulay.
We give a unified formulation of a mass for arbitrary abelian varieties with PEL-structures and show that it equals a weighted class number of a reductive $\Q$-group $G$ relative to an open compact subgroup $U$ of $G(\A_f)$, or simply called an {\it arithmetic mass}. We classify the special objects for which our formulation remains valid over algebraic closed fields. As a result, we show that the set of basic points in a mod $p$ moduli space of PEL-type with a local condition (and a mild condition subject to the Hasse principle) can be expressed as a double coset space and its mass equals an a…
The motivic Hecke algebra for PEL Shimura varieties
本研究构建了作用于PEL志村簇上同调的动机Hecke代数，定义其为动机三角范畴中可构造动机自同态代数的子代数，结合内部动机理论得到带该代数作用的Chow动机，并列出适用情形。
We show, using the trace formula, that any Newton stratum of a Shimura variety of PEL-type of types (A) and (C) is non-empty at the primes of good reduction. Furthermore we prove conditionally the non-emptiness for Shimura data associated to odd Spin groups. Our results are conditional on Rapoport-Langlands conjecture and Arthur's conjectures on the discrete spectrum. Both these results have been announced by Arthur and Kisin in significant cases.
In this paper we study the reduction of PEL-Shimura varieties associated to unitary groups of signature (n-1,1) in the inert and unramified case. We describe the Newton polygon and the Ekedahl-Oort stratification. We further study the associated moduli space of isogenies and use a variant of the local model to give a generic description. As an application we prove a conjecture of Blasius und Rogawski about the congruence relation if the integer n is even.
Generalising the recent method of Andreatta, Iovita, and Pilloni for cuspidal forms, we construct an eigenvariety for symplectic and unitary groups that parametrises systems of eigenvalues of overconvergent and locally analytic $p$-adic automorphic forms. This is achieved by gluing some intermediates eigenvarieties of a fixed 'degree of cuspidality'. The dimension of these eigenvarieties is explicit and depends on the degree of cuspidality, it is maximal for cuspidal forms and it is $1$ for forms that are 'not cuspidal at all'. Under mild assumption, we are able to prove a conjecture of Urban …
In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink space whose special fibre is described by the Bruhat-Tits stratification. The description for this supersingular locus in the case where the prime is inert in imaginary quadratic field is already known to Howard and Pappas by exploiting the exceptional isomorphism. Our method is more direct without using the exceptional isomorphism.
Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…
Cycles on splitting models of Shimura varieties
本文通过分裂模型构造了 Shimura 变种的 exotic Hecke 对应关系，并验证了 Tate 猜想的通用实例。
Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$
本文研究了特征标为(1,n-1)的基本无分歧PEL单位aryRapoport-Zink空间在最大水平下的上同调，通过谱序列和p进统一化方法，证明了Frobenius作用的半单性，并在n=3或4时描述了超奇异位置的上同调。
Nearby cycles on the local model for the $\mathrm{GU}(n-1,1)$ PEL Shimura variety over a ramified prime
本文计算了GU(n-1,1) PEL Shimura 品在分圆素数上的局部模型的ℓ-adic附近层的上同调sheaf，通过研究局部模型的奇异点以及吹胀后的半稳定降解，证明当n为奇数时附近层是平凡的，而当n为偶数时仅有一个高上同调sheaf不消失，并计算了对应的Frobenius特征值。
Let k be a perfect field of characteristic p>0. We prove the existence of ascending and descending slope filtrations for Shimura p-divisible objects over k. We use them to classify rationally these objects over \bar k. Among geometric applications, we mention two. First we formulate Manin problems for Shimura varieties of Hodge type. We solve them if either p\Ge 3 or p=2 and two mild conditions hold. Second we formulate integral Manin problems. We solve them for certain Shimura varieties of PEL type.
In this article, we generalize the work of H.Hida and V.Pilloni to construct $p$-adic families of $μ$-ordinary modular forms on Shimura varieties of Hodge type $Sh(G,X)$ associated to a Shimura datum $(G,X)$ where $G$ is a connected reductive group over $\mathbb{Q}$ and is unramified at $p$, such that the adjoint quotient $G^\mathrm{ad}$ has no simple factors isomorphic to $\mathrm{PGL}_2$.
We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set…
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.
Under simplifying hypotheses we prove a relation between the l-adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields. We obtain our results using the trace formula and truncation of the formula of Kottwitz for the number of points on a Shimura variety over a finite fields.
Integral canonical models for Spin Shimura varieties
本文为附加于Spin群的Shimura簇在奇素数处构造了正则整典范模型并证明其典范性，同时给出了良好紧化，结果应用于K3曲面的Tate猜想及Kudla计划。
The Kottwitz conjecture describes the cohomology of basic Rapoport-Zink spaces using local Langlands correspondences. In this paper, via geometrical studies of some Kottwitz-type Shimura varieties, we prove this conjecture for basic simple unramified unitary PEL type Rapoport-Zink spaces of signature (1, n-1).
We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" $p$-adic integral models of these Shimura varieties and study their 'etale local structure. In this 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 in our previous paper math.AG/0607130.
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…
This work is the first part in a series of three dedicated to the foundations of integral aspects of Shimura varieties and of Fontaine's categories. It deals mostly with the unramified context of (arbitrary) mixed characteristic (0,p). Among the topics covered we mention: the generalization of the classical Serre-Tate theory of ordinary p-divisible groups and of their canonical lifts, the generalization of the classical Serre-Tate-Dwork-Katz theory of (crystalline) coordinates for ordinary abelian varieties, the strong form of the generalized Manin problem, global deformations in the generaliz…
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.
Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{ρ_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2iπ/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimur…
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.
Igusa Stacks and the Cohomology of Shimura Varieties
本文针对所有Hodge型志村簇构造函子性Igusa叠，依托该构造及Fargues-Scholze谱作用证明其与半单局部朗兰兹对应的相容性，还在紧合情形得到新的上同调挠消失结果。
Torsion Vanishing for Some Shimura Varieties
本文推广了扭系数消失结果，结合多种技术方法，建立了新的上同调描述，并提出更一般的扭系数消失猜想。
Integral canonical models of exceptional Shimura varieties
本文证明例外 Shimura 簇在足够大的素数处存在积分典范模型，并推广至周期映射像，进而得到若干算术推论。
We construct a generalization of the Hasse invariant for any Shimura variety of PEL type A over a prime of good reduction, whose vanishing locus is the open and dense μ-ordinary locus.
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