我们证明了关于酉相似群的 Shimura 曲线的 p-adic 一致化，该群与相对于任意极大全实子域为 F 的 CM 域 K 的特定二元斜厄米特空间 V 有关。对于 F 的一个在 K 中不分裂且使得 V_v 为各向异性的素除子 v|p，设 ν 为 v 到反射域 E 的一个延拓。我们通过带有合适极化及与 p 互素的水平结构的阿贝尔概型的模问题，定义了在 Spec O_{E,(ν)} 上相应 Shimura 曲线的积分模型。该模问题的表述涉及 Kottwitz 条件、Eisenstein 条件以及一个调整不变量。前两个条件是关于阿贝尔簇李代数的条件；最后一个条件是关于极化 Riemann 形式的条件。该模型沿其特殊纤维的形式完备化的一致化由 F_v 的形式 Drinfeld 上半平面给出。证明依赖于收缩函子的构造，该函子将严格形式 O_{F_v}-模的相对 Rapoport-Zink 空间与由模问题产生的 p-可除群的 Rapoport-Zink 空间联系起来，其中当 F_v≠Q_p 时，O_{F_v}-作用通常不是严格的。我们的主要工具是 display 理论，特别是 Ahsendorf 函子。
英文摘要：
We prove $p$-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew hermitian spaces $V$ with respect to an arbitrary CM field $K$ with maximal totally real subfield $F$. For a place $v|p$ of $F$ that is not split in $K$ and for which $V_v$ is anisotropic, let $ν$ be an extension of $v$ to the reflex field $E$. We define an integral model of the corresponding Shimura curve over ${\rm Spec}\, O_{E, (ν)}$ by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to $p$. The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The first two conditions are conditions on the Lie algebra of the abelian varieties; the last condition is a condition on the Riemann form of the polarization. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane for $F_v$. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal $O_{F_v}$-modules with a Rapoport-Zink space of $p$-divisible groups which arise from the moduli problem, where the $O_{F_v}$-action is usually not strict when $F_v\ne \mathbb {Q}_p$. Our main tool is the theory of displays, in particular the Ahsendorf functor.
In this paper we generalize Cherednik's method and prove that certain Shimura varieties corresponding to groups of unitary similitudes and automorphic vector bundles over them have p-adic uniformization. This is proved for Shimura varieties, uniformized by the complex unit ball, when the central simple algebra over a CM-field defining the group of unitary similitudes has Brauer invariant 1/d at p. In this case, Shimura varieties can be uniformized by Drinfeld's covers of p-adic upper half-spaces.
Let D be a quaternion division algebra over a totally real number field F which splits exactly at one infinite place. We assume that there is a p-adic place where D doesn't split. Then the associated Shimura curve has a Cherednik unifomization by the p-adic upper half plane. We define an integral model of the Shimura curve an over the integers O of this p-adic place. We prove an uniformization theorem for the completion of the model over the formal spectrum Spf O.
The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over the field of rational numbers at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of Cerednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenberg's construction of local points on …
In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product (of equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [V], and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety…
We establish a relation between intersection numbers of special cycles on a Shimura curve and special values of derivatives of metaplectic Eisenstein series at a place of bad reduction where p-adic uniformization in the sense of Cherednik and Drinfeld holds. The result extends the one established by one of us (S. Kudla: Ann. of Math. 146 (1997)) for the archimedean place and for the non-archimedean places of good reduction. The bulk of the paper is concerned with the corresponding problem on the Drinfeld upper half plane (the formal scheme version).
We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of…
Projective structures on compact real manifolds are classical objects in real differential geometry. Complex manifolds with a holomorphic projective structure on the other hand form a special class as soon as the dimension is greater than one. In the Kähler Einstein case, projective space, tori and ball quotients are essentially the only examples. They can be described purely in terms of Chern class conditions. We give a complete classification of all projective manifolds carrying a projective structure. The only additional examples are modular abelian families over quaternionic Shimura curves…
We prove a Cherednik style $p$-adic uniformization theorem for Shimura varieties associated to certain groups of unitary similitudes of size two over totally real fields. Our basic tool is the alternative modular interpretation of the Drinfeld $p$-adic halfplane of our earlier paper (arXiv 1108.5713)
We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.
This is a survey article that advertizes the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also review their theory in the light of this idea. We also discuss conjectures on the $\ell$-adic cohomology of local Shimura varieties.
In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization, in terms of geometry mod p, of curves in positive characteristics which are reduction of Shimura curves over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian varieties over k, char k=p. We show that some conditions on the l-adic monodromy over such a curve imply that this curve can be lifted to a Shimura curve.
In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, th…
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…
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V…
We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3(Q2) that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake plane yet distinct from it and the other fake planes that arise from 2-adic uniformization by torsion-free groups. As part of the proof, and of independent interest, we compute the homotopy type of the Berkovich space of our plane.
Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of $\mathbb{Q}_p$, $K$ the maximal unramified extension of $\mathbb{Q}_p$, $\overline{K}$ some fixed algebraic closure of $K$, and $\mathbb{C}_p$ the completion of $\overline{K}$. Let $G_F$ the absolute Galois group of $F$. Let $A$ be an abelian variety defined over $F$, with good reduction. Classically, the Fontaine integral was seen as a Hodge--Tate comparison morphism, i.e. as a map $φ_{A} \otimes 1_{\mathbb{C}_p}\colon T_p(A)\otimes_{\mathbb{Z}_p}\mathbb{C}_p\to \text{Lie}(A)(F)\otimes_F\mathbb{C}_p(1)$, and as such…
We prove that all $p$-adic period domains (and their non-minuscule analogues) are geometrically connected. This answers a question of Hartl [Har13] and has interesting consequences to the geometry of Shimura and local Shimura varieties.
The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in…
In this paper, we define and construct canonical filtered $F$-crystals with $G$-structure over the integral models for Shimura varieties of abelian type at hyperspecial level defined by Kisin. We check that these are related by $p$-adic comparison theorems to the usual lisse sheaves, and as an application we also use this to show that the Galois representations generated from the $p$-adic étale cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline, at least in the case of proper abelian type Shimura varieties.
We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image contains the $\mathbb{Z}_\ell$-points coming from the simply connected cover of the derived subgroup of $G$. This can be regarded as a geometric version of the integral $\ell$-adic Mumford-Tate conjecture.
We consider a finite étale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton $Γ_f=(Γ_Y,Γ_X)$ of the morphism $f$. We prove that $Γ_f$ radializes $f$ if and only if $Γ_X$ controls the pushforward of the constant $p$-adic differential equation $f_*(\mathcal{O}_Y,d_Y)$. Furthermore, when $f$ is a finite étale morphism of open unit discs, we prove that $f$ is radial if and only if the number of preimages of a point $x\in X$, counted without multiplicity, only depe…
Let f be a modular form of weight k>=2 and level N, let K be a quadratic imaginary field, and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level and the field K, one can attach to this data a p-adic L-function L_p(f,K,s), as done by Bertolini-Darmon-Iovita-Spiess. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,...,k-1, and therefore one is interested in the values of its derivative in this range. We construct, for k>=4, a Chow motive endowed with a distinguished collection of a…
Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the curve to the ring $W(k)$ of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over $X/W$. As a consequence, it gives rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group $π_1(X_K)$ of the generic fiber of $X$. This curve $X$ and its associated representation is in close relation wit…
We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.
We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the group U(n, n). This is done by relating the arithmetic cycles to their formal counterpart from Part I via non-archimedean uniformization, and by relating the Fourier coeff…
We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is greater than or equal to 2, $\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\Aut (X_0(D, N)) = W_0(D, N)$ except for finitely many values of $(D, N)$ and we provide criteria that allow us to show that this is indeed often the case. Our met…
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the μ-ordinary locus, when applied to scalar-valued forms.
We give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\ZZ[\z_7]$.
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.
In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-s…
Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the $l$-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport-Zink spaces which are Hodge type and Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink space into an appropriate space of EL type, for which th…
Let (P,X) be Shimura data, M=M(P,X,K) the Shimura variety of level K. To an algebraic representation of P, one can associate a mixed sheaf (variation of Hodge structure, l-adic sheaf) on M. In the paper, we study the degeneration of such sheaves along strata in toroidal compactifications of M. The main result (2.8 in the Hodge setting, 3.9 in the l-adic setting) gives a formula for this degeneration in terms of Hochschild cohomology of certain unipotent subgroups of P. The new version differs from the earlier one in that the proof of 2.8 was rewritten. In particular, the effect of Saito's spec…
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 interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p-adic weight shifted by 2. Our construction is purely geometric, using Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves, and should thus generalize to higher rank groups.
We show that the moduli space of ordered 5 points on the projective line is isomorphic to an arithmetic quotient of a complex ball by using the theory of periods of K3 surfaces. We also discuss a relation between our uniformization and the one given by Shimura, Terada and Deligne-Mostow.
