Plain TeX, 44 pages. This is a completely revised version of our March 2002 preprint; part of the results are now given in a separate paper (math.AG/0208161)
The main goal of this paper is to generalize Serre-Tate theory of "ordinary" local moduli to Shimura varieties of PEL type. To this end we develop a generalized notion of ordinariness, we prove a number of basic results about this, and we study the formal deformations of ordinary objects. In general, the formal deformation spaces get a new "group-like" structure that we call a "cascade". Further the paper contains some results on the Ekedahl-Oort stratification of PEL moduli spaces, and an application of our results to the study of congruence relations.
We study a generalization of Serre--Tate theory of ordinary abelian varieties and their deformation spaces. This generalization deals with abelian varieties equipped with additional structures. The additional structures can be not only an action of a semisimple algebra and a polarization, but more generally the data given by some ``crystalline Hodge cycles'' (a $p$-adic version of a Hodge cycle in the sense of motives). Compared to Serre--Tate ordinary theory, new phenomena appear in this generalized context. We give an application of this theory to the existence of ``good'' integral models of…
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under…
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…
This paper concerns the Algebraic Sato--Tate and Sato--Tate conjectures, based on Serre's original motivic formulation, with an eye towards explicit computations of Sato--Tate groups. We build on the algebraic framework for the Sato--Tate conjecture introduced in a previous paper, which used Deligne's motivic category for absolute Hodge cycles and was restricted to motives of odd weight. Here, we allow general weight and some other motivic categories, notably André's motivic category of motivated cycles; moreover, some results are also new in the odd weight case. The paper consists of two part…
We study the formal neighbourhood of a point in $μ$-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a shifted cascade. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms in analogy with the Serre-Tate canonical lift of an ordinary abelian variety.
Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without non…
In this paper we study various aspects of the Ekedahl-Serre problem. We formulate questions of Ekedahl-Serre type and Coleman-Oort type for general weakly special subvarieties in the Siegel moduli space, propose a conjecture relating these two questions, and provide examples supporting these questions. The main new result is an upper bound of genera for curves over number fields whose Jacobians are isogeneous to products of elliptic curves satisfying the Sato-Tate equidistribution, and we also refine previous results showing that certain weakly special subvarieties only meet the open Torelli l…
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.
In this paper we study the Newton stratification on the reduction of Shimura varieties of PEL type with hyperspecial level structure and the Newton stratification on the deformation space of a Barsotti-Tate group with PEL structure. Our main result is a formula for the dimension of Newton strata and the description their closure in each of the two cases. Furthermore, we calculate the dimension of some Rapoport-Zink spaces as an intermediate result.
In this paper, we construct the almost product structure of the minimal Newton stratum in deformation spaces of Barsotti-Tate groups with crystalline Tate tensors, similar to Oort's and Mantovan's construction for Shimura varieties of PEL-type. It allows us to describe the geometry of the Newton stratum in terms of the geometry of two simpler objects, the central leaf and the isogeny leaf. This yields the dimension and the closure relations of the Newton strata in the deformation space. In particular, their nonemptiness shows that a generalisation of Grothendieck's conjecture of deformations o…
We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^*$-algebra $\mathcal{A}_X$ of a projective variety $X$ as the norm-closure of a representation of the twisted homogeneous coordinate ring of $X$ by the linear operators on a Hilbert space $\mathcal{H}$. It is proved that $X$ is homeomorphic to the space of all irreducible representations of the crossed product of $\mathcal{A}_X$ by an automorphism of $\mathcal{A}_X$. The case of rational elliptic curves $X$ is considered in detail.
According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives.
We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mézard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially …
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.
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 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 consider p-divisible groups (also called Barsotti-Tate groups) in characteristic p, their deformations, and we draw some conclusions. For such a group we can define its Newton polygon (abbreviated NP). This is invariant under isogeny. For an abelian variety (in characteristic p) the Newton polygon of its p-divisible group is ``symmetric''. In 1963 Manin conjectured that conversely any symmetric Newton polygon is ``algebroid''; i.e., it is the Newton polygon of an abelian variety. This conjecture was shown to be true and was proved with the help of the ``Honda-Serre-Tate theory''. We give an…
Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. When $A$ is isogenous to a product of simple abelian varieties of $\GSp$ type, i.e. whose Mumford-Tate group is "generic" (isomorphic to the group of symplectic similitudes) and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of $A$. The result is unconditional for a product of simple abelian varieties with endomorph…
Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three conditions, we compute the optimal exponent for this bound, in terms of the dimension of abelian subvarieties and their endomorphism rings. The three hypothesis are the following: $(1)$ $A$ is geometrically isogenous to a product of simple abelian varieties of type I, II or III, according to the Albert classification; $(2)$ $A$ is of "Lefschetz type", that is, t…
A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystal…
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…
