Eva Viehmann
We study the global structure of moduli spaces of quasi-isogenies of p-divisible groups introduced by Rapoport and Zink. We determine their dimensions and their sets of connected components and of irreducible components. If the isocrystals of the p-divisible groups are simple, we compute the cohomology of the moduli space. As an application we determine which moduli spaces are smooth.
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We study the global structure of moduli spaces of quasi-isogenies of polarized p-divisible groups introduced by Rapoport and Zink. Using the corresponding results for non-polarized p-divisible groups from a previous paper, we determine their dimensions and their sets of connected components and of irreducible components.
Lecture notes at a conference on Arithmetic Geometry, Goettingen, July/August 2006: Density of ordinary Hecke orbits and a conjecture by Grothendieck on deformations of p-divisible groups.
The moduli space of stable maps with divisible ramification uses $r$-th roots of a canonical ramification section to parametrise stable maps whose ramification orders are divisible by a fixed integer $r$. In this article, a virtual fundamental class is constructed while letting domain curves have a positive genus; hence removing the restriction of the domain curves being genus zero. We apply the techniques of virtual localisation and obtain the $r$-ELSV formula as an intersection of the virtual class with a pullback via the branch morphism.
Let $O_D$ be the ring of integers in a division algebra of invariant $1/n$ over a p-adic local field. Drinfeld proved that the moduli problem of special formal $O_D$-modules is representable by Deligne's formal scheme version of the Drinfeld p-adic halfspace. In this paper we exhibit other moduli spaces of formal $p$-divisible groups which are represented by $p$-adic formal schemes whose generic fibers are isomorphic to the Drinfeld p-adic halfspace. We also prove an analogue concerning the Lubin-Tate moduli space.
The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.
We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p-adic Hodge theory. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of p-divisible groups over the ring of integers of a complete algebraically closed fi…
We develop a theory for stable maps to curves with divisible ramification. For a fixed integer $r>0$, we show that the condition of every ramification locus being divisible by $r$ is equivalent to the existence of an $r$th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to $r$-s…
This paper studies deformations and birational maps between singular moduli spaces of semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that under certain conditions, two such moduli spaces of the same dimension can be connected by deformations and birational maps.
Let k be a perfect field of characteristic p > 2. In this note, we show that the local moduli space of a non-supersingular K3 surface over k with trivial deformation of the associated enlarged formal Brauer group admits a natural p-divisible formal group structure.
Slope filtrations of log $p$-divisible groups
本文研究了 log p-可除群的斜过滤，推广了 Oort-Zink 关于 p-可除群的同源性结果。
A criterion is given which assures that two p-divisible groups X and Y over an algebraically closed field of characteristic p are isomorphic when their p-kernels X[p] and Y[p] are isomorphic.
We shall study minimal complex surfaces with $c^2 = 9$ and $χ=5$ whose canonical classes are divisible by $3$ in the integral cohomology groups, where $c_1^2$ and $χ$ denote the first Chern number of an algebraic surface and the Euler characteristic of the structure sheaf, respectively. The main results are a structure theorem for such surfaces, the unirationality of the moduli space, and a description of the behavior of the canonical map. As a byproduct, we shall also rule out a certain case mentioned in a paper by Ciliberto--Francia--Mendes Lopes. Since the irregularity $q$ vanishes for our …
A p-divisible group over a base scheme in characteristic p in general does not admit a slope filtration. Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S; we prove that there exists an isogeny from X to Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim(S) = 1 and by T. Zink for dim(S) > 0. We give an example of a p-divisible group over a non-normal base which does not admit an isogeny to a p-divisible group with a slope filtration.
We study invariant functions on the reductions mod p^n of p-divisible groups. The proof of the main result, which applies to one-dimensional groups, combines results of Tate with van der Put's solution of his p-adic Corona problem. For higher dimensional groups a generalization of the p-adic Corona problem would have to be solved.
We notice that for $0<d\le 6$ the Poincaré polynomial of Simpson moduli space $M_{dm + 1}(\mathbb P_2)$ is divisible by the Poincaré polynomial of the projective space $\mathbb P_{3d-1}$. A somehow regular behaviour of the difference of the Poincaré polynomials of the Hilbert scheme of $\frac{(d-2)(d-1)}{2}$ points on $\mathbb P_2$ and the moduli space of Kronecker modules $N(3; d-2, d-1)$ is noticed for $d=4, 5, 6$.
In this paper, for any pair $(ζ, ξ)$ of Newton polygons with $ζ\prec ξ$, we construct a concrete specialization from the minimal $p$-divisible group of $ξ$ to the minimal $p$-divisible group of $ζ$ by a beautiful induction. This in particular gives the affirmative answer to the unpolarized analogue of a question by Oort on the boundaries of central streams, and gives another proof of the dimension formula of the central leaves in the unpolarized case.
We construct a compactification M_d of the moduli space of plane curves of degree d. We regard a plane curve C as a surface-divisor pair (P^2,C) and define M_d as a moduli space of pairs (X,D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack M_d is smooth and the degenerate surfaces X can be described explicitly.
A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons.
A Dieudonné theory for analytic p-divisible groups and applications to Shimura varieties
本文研究了解析p-可除群的Dieudonné理论，并将其应用于Shimura品种，证明了与Hodge-Tate三元组的等价性，并重新解释了Hodge-Tate周期映射。
The notions Hodge-Newton decomposition and Hodge-Newton filtration for F-crystals are due to Katz and generalize Messing's result on the existence of the local-étale filtration for p-divisible groups. Recently, some of Katz's classical results have been generalized by Kottwitz to the context of F-crystals with additional structures and by Moonen to $μ$-ordinary p-divisible groups. In this paper, we discuss further generalizations to the situation of crystals in characteristic p and of p-divisible groups with additional structure by endomorphisms.
F-isocrystals of Higher Direct Images of $p$-Divisible Groups
本文研究特征p完美域上有限生成域k的光滑射影簇上p-可除群的高阶直像，证明其形式群与p-可除群同源，且可除部分的Dieudonné晶体与对应F-等晶的斜率[0,1]部分典范同构，解答了Artin–Mazur相关问题的有理形式。
An algebraicity conjecture of Drinfeld and the moduli of $p$-divisible groups
本文利用德拉姆和布特尔-帕帕斯发展的新技术，统一构造了与p-可除群相关的模，验证了德拉姆的猜想，并推广了相关结果。
We investigate qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.
We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces $\mathcal M_{En,h}^a$of polarized Enriques surfaces with fixed polarization type $h$ to the moduli space $\mathcal F_g$ of polarized $K3$ surfaces of genus $g$ with $g=h^2+1$, and we exhibit a naturally defined locus $Σ_g\subset\mathcal F_g$. One direct consequence of the Borisov-Nuer conjecture is that $Σ_g$ would be contained in a particular Noether-Lefschetz divisor in $\mathcal F_g$, which we call the Borisov-Nuer divisor and we denote by $\mathcal{BN}_g$. In this short note, we prove that $…
In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.
Oort's conjecture on automorphisms of generic supersingular abelian varieties
本文证明了Oort猜想：在特征p的亏格g主极化阿贝尔簇模空间的超奇异轨迹上，除g=2或3且p=2的情况外，通用主极化阿贝尔簇的自同构群仅为±1。
Consider all moduli points corresponding with polarized abelian varieties in characteristic p such that the associated quasi-polarized p-divisible group is geometrically isomorphic with a given one. This defines a subset C of the moduli space. We show that C is closed in the corresponding open Newton polygon stratum. In this way the stratum is the disjoint union of such "central leaves". Under local-local isogenies we consider the orbit of one point: we obtain a second type of leaves. Every stratum, up to a finite map, is the product of one of its central leaves and one of its isogeny leaves. …
Moduli spaces of nilpotent displays
该研究利用Artin判别法与截断display理论，在不借助p-可除群的情况下证明了幂零display拟同构变形模问题的可表性，还给出了Rapoport-Zink空间可表性的新证法。
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$ which is not isoclinic. Let $\scrD$ (resp. $\scrD_k$) be the formal deformation space of $D$ over $\Spf(W(k))$ (resp. over $\Spf(k)$). We use axioms to construct formal subschemes $\scrG_k$ of $\scrD_k$ that: (i) have canonical structures of formal Lie groups over $\Spf(k)$ associated to $p$-divisible groups over $k$, and (ii) give birth, via all geometric points $\Spf(K)\to\scrG_k$, to $p$-divisible groups over $K$ that are isomorphic to $D_K$. We also identify when there exist formal s…
Moduli spaces of bounded local $G$-shtukas are a group-theoretic generalization of the function field analog of Rapoport and Zink's moduli spaces of $p$-divisible groups. In this article we generalize some very prominent concepts in the theory of Rapoport-Zink spaces to our setting. More precisely, we define period spaces, as well as the period map from a moduli space of bounded local $G$-shtukas to the corresponding period space, and we determine the image of the period map. Furthermore, we define a tower of coverings of the generic fiber of the moduli space which is equipped with a Hecke act…
Sheared displays and $p$-divisible groups
利用剪切Witt向量为$p$-可除群建立Dieudonné理论。
Let $K$ be a $p$-adic field. We continue to develop the theory of rigid analytic $p$-divisible groups over $K$. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic $p$-divisible groups "in finite level" without perfectoid spaces. We then establish some results about families of rigid analytic $p$-divisible groups. This allows us to prove a "minimality" result in the sense of birationnal geometry for integral models of unramified Rapoport-Zink spaces.
We extend the results of Kottwitz on points of Shimura varieties over finite fields to cases of bad reduction. The "test function" whose twisted orbital integrals appear in the final expression is defined geometrically using deformation spaces of p-divisible groups.
We introduce the notion of primitive elements in arbitrary truncated $p$-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the "points of exact order $N$," developed by Drinfeld and Katz-Mazur for elliptic curves.
Assume that $p>2$, and let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring with residue field admitting a finite $p$-basis, and let $R$ be a formally smooth formally finite-type $\mathscr{O}_K$-algebra. (Indeed, we allow slightly more general rings $R$.) We construct an anti-equivalence of categories between the categories of $p$-divisible groups over $R$ and certain semi-linear algebra objects which generalise $(φ,\mathfrak{S})$-modules of height $\leqslant1$ (or Kisin modules). A similar classification result for $p$-power order finite flat group schemes is deduced from the classificat…
We give a criterion which allows to determine, in terms of the combinatorics of the root system of the general linear group, which p-kernels occur in an isogeny class of p-divisible groups over an algebraically closed field of positive characteristic. As an application we obtain a criterion for the non-emptiness of certain affine Deligne-Lusztig varieties associated to the general linear group.
We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $μ$ ; this condition consists in a filtration on the sheaf of differentials $ω_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying …
We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, °L)$ unobstructed components. In addition, if $k$ is divisible by $r_1(r-r_1)(g-1)$ for $1\le r_1\le r-1$, there is an additional obstructed component of the expected dimension for each such $r_1$. We construct families of obstructed components and show that their generic point is not the generic vector…
Starting from our work on Harder-Narasimhan filtrations of finite flat group schemes over a $p$-adic field, we developp a theory of Harder-Narasimhan filtrations for $p$-divisible groups. We apply this to the study of the geometry of period morphisms for Rapoport-Zink spaces and to the $p$-adic geometry of Shimura varieties. We define and study in particular some fundamental domains for the action of Hecke correspondences.
Log p-divisible groups associated to log 1-motives
该研究证明了对数p-可除群的Kato分类定理，定义了标准扩张概念并证明对数p-可除群形式对数光滑，还研究了对数1-motive的相关群结构并证明对数阿贝尔簇的Serre-Tate定理。
Invariant Functions on $p$-divisible Groups and the $p$-adic Corona Problem II
本注记消除了同名论文主要结果的一维限制，拓展了p-可除群与p进 corona 问题相关研究的适用范围。
Bounded local G-shtukas are function field analogs for p-divisible groups with extra structure. We describe their deformations and moduli spaces. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine Deligne-Lusztig varieties. For basic Newton polygons the closed Newton stratum in the universal deformation of a local G-shtuka is isomorphic to the completion of a corresponding affine Deligne-Lusztig variety in that point. This yields bounds on the dimension and proves equidimensionality of the basic affine Deligne-Luszt…
Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k is included in \bar{F}_p (and p>2), we then use this new classification to prove that any crystalline representation of t…
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latte…
Let k be an algebraically closed field of characteristic p>0. Let H be a supersingular p-divisible group over k of height 2d. We show that H is uniquely determined up to isomorphism by its truncation of level d (i.e., by H[p^d]). This proves Traverso's truncation conjecture for supersingular p-divisible groups. If H has a principal quasi-polarization λ, we show that (H,λ) is also uniquely determined up to isomorphism by its principally quasi-polarized truncated Barsotti--Tate group of level d (i.e., by (H[p^d],λ[p^d])).
We prove that, for any $g \geq 2$, the étale double cover $ρ_g:\mathcal{E}_{g} \to \widehat{\mathcal{E}}_{g}$ from the moduli space $\mathcal{E}_{g}$ of complex polarized genus $g$ Enriques surfaces to the moduli space $\widehat{\mathcal{E}}_{g}$ of numerically polarized genus $g$ Enriques surfaces is disconnected precisely over irreducible components of $\widehat{\mathcal{E}}_{g}$ parametrizing $2$-divisible classes, answering a question of Gritsenko and Hulek. We characterize all irreducible components of $\mathcal{E}_{g}$ in terms of a new invariant of line bundles on Enriques surfaces that…
A $p$-divisible group over a field $K$ admits a slope decomposition; associated to each slope $λ$ is an integer $m$ and a representation $\gal(K) \ra \gl_m(D_λ)$, where $D_λ$ is the $\rat_p$-division algebra with Brauer invariant $[λ]$. We call $m$ the multiplicity of $λ$ in the $p$-divisible group. Let $G_0$ be a $p$-divisible group over a field $k$. Suppose that $λ$ is not a slope of $G_0$, but that there exists a deformation of $G$ in which $λ$ appears with multiplicity one. Assume that $λ\not= (s-1)/s$ for any natural number $s>1$. We show that there exists a deformation $G/R$ of $G_0/k$ s…
Traverso's Isogeny Conjecture for Some Unitary p-Divisible Groups
本文研究签名 (a,b) 的超奇异酉 p-可除群，完整刻画其最小高度，给出同源截断界，并用酉 Shimura 簇的 BT_m 分层重述结果。
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d\in\dbN$. Let $b_{c,d}\ge 1$ be the smallest integer such that for any two $p$-divisible groups $H$ and $H^\prime$ over $k$ of codimension $c$ and dimension $d$ the following assertion holds: If $H[p^{b_{c,d}}]$ and $H^\prime[p^{b_{c,d}}]$ are isomorphic, then $H$ and $H^\prime$ are isogenous. We show that $b_{c,d}=\lceil{cd\over {c+d}}\rceil$. This proves Traverso's isogeny conjecture for $p$-divisible groups over $k$.
Log $p$-divisible groups associated with semi-abelian degeneration
本文研究阿贝尔概形沿正规交叉除子半阿贝尔退化时挠点群概形的退化行为，证明其退化为对数有限群概形，并说明该对象比半阿贝尔概形挠点的拟有限平坦群概形保留更多信息。
Let $\mathscr{O}_K$ be a 2-adic discrete valuation ring with perfect residue field $k$. We classify $p$-divisible groups and $p$-power order finite flat group schemes over $\mathscr{O}_K$ in terms of certain Frobenius module over $\mathfrak{S}:=W(k)[[u]]$. We also show the compatibility with crystalline Dieudonné theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, and we additionally obtain the the compatibility with associated Galois representations.
In this paper we study the slope stratification on the good reduction of the type C family Shimura varieties. We show that there is an open dense subset $U$ of the moduli space such that any point in $U$ can be deformed to a point with a given lower {\it admissible} Newton polygon. For the Siegel moduli spaces, this is obtained by F. Oort which plays an important role in his proof of the strong Grothendieck conjecture concerning the slope stratification. We also investigate the $p$-divisible groups and their isogeny classes arising from the abelian varieties in question.
In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.
Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields
本文提出了一种计算有限域上曲线的p-可除群的算法，基于晶体上同调的Frobenius和Verschiebung算子，改进了Tuitman的p进点计数算法。
Let $ S $ be the special fibre of the good reduction of a Shimura variety of Hodge type. By constructing adapted deformations for the associated $p$-divisible groups of $ S $, we manage to construct a morphism from $S$ to some quotient sheaf of the loop group associated with $S$. We show that the geometric fibres of this morphism give back the Ekedahl-Oort strata of $S$. For any geometric point $ x $ of $ S $, we give a deformation over $ W(k(x)) $ of the $ p $-divisible group associated with $ x $ by (non-canonically) constructing a Breuil-Kisin window (which corresponds to a $ p $-divisible …
More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case
本文针对 p 进域二次分歧扩张上的酉群，构造了带仿射仿射层次的正则形式模空间及其间的对应，提出算术转移猜想，并在最低维情形下给出证明。
Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.
We develop the analog of crystalline Dieudonné theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of …
Hodge-Newton filtration for $p$-divisible groups with ramified endomorphism structure
本文基于有限平坦群概型的 Harder-Narasimhan 理论，证明了具有分歧自同态结构的 $p$-可分群上 Hodge-Newton 滤波的存在性，并给出了其存在的充分条件。
Log prismatic Dieudonné theory for log $p$-divisible groups over $\mathcal{O}_{K}$
本文证明了赋予典范对数结构的完备离散赋值环上的对数 $p$-可除群，对应于绝对对数棱镜景上的 Dieudonné 晶体，并利用对数下降将其归约为经典棱镜对应。
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