Barsotti-Tate Group Research Articles

Purity for Barsotti–Tate groups in some mixed characteristic situations

Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of $k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $\text{Frac}(R)$ which extends to every local ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $e\le p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.

On Deformations of 1-motives

AbstractAccording 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.

Normalization of Closed Ekedahl—Oort Strata

AbstractWe apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

The almost product structure of Newton strata in the Deformation space of a Barsotti–Tate group with crystalline Tate tensors

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 of Barsotti-Tate groups with given Newton polygon holds. As an application, we determine analogous geometric properties of the Newton stratification of Shimura varieties of Hodge type and prove the equidimensionality of Rapoport-Zink spaces of Hodge type.

The geometry of Newton strata in the reduction modulo p 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.

THE DISTRIBUTION OF TORSION SUBSCHEMES OF ABELIAN VARIETIES

We consider the distribution of $p$-power group schemes among the torsion of abelian varieties over finite fields of characteristic $p$, as follows. Fix natural numbers $g$ and $n$, and let ${\it\xi}$ be a non-supersingular principally quasipolarized Barsotti–Tate group of level $n$. We classify the $\mathbb{F}_{q}$-rational forms ${\it\xi}^{{\it\alpha}}$ of ${\it\xi}$. Among all principally polarized abelian varieties $X/\mathbb{F}_{q}$ of dimension $g$ with $X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$, we compute the frequency with which $X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$. The error in our estimate is bounded by $D/\sqrt{q}$, where $D$ depends on $g$, $n$, and $p$, but not on ${\it\xi}$.

On lower ramification subgroups and canonical subgroups

Let p be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fraction field of the Witt ring of k. Let G be a finite flat commutative group scheme over O_K killed by some p-power. In this paper, we prove a description of ramification subgroups of G via the Breuil-Kisin classification, generalizing the author's previous result on the case where G is killed by p>2. As an application, we also prove that the higher canonical subgroup of a level n truncated Barsotti-Tate group G over O_K coincides with lower ramification subgroups of G if the Hodge height of G is less than (p-1)/p^n.

Canonical subgroups via Breuil–Kisin modules for [formula omitted

Let p be a rational prime and K/Qp be an extension of complete discrete valuation fields. Let G be a truncated Barsotti–Tate group of level n, height h and dimension d over OK with 0<d<h. In this paper, we prove the existence of higher canonical subgroups for G with standard properties if the Hodge height of G is less than 1/(pn−2(p+1)), including the case of p=2.

Generalization of Deuring reduction theorem

In this paper we generalize the Deuring theorem on a reduction of an elliptic curve with complex multiplication. More precisely, for an Abelian variety A, arising after reduction of an Abelian variety with complex multiplication by a CM field K over a number field at a place of good reduction. We establish a connection between a decomposition of the first truncated Barsotti–Tate group scheme A[p] and a decomposition of pOK into prime ideals. In particular, we produce these explicit relationships for Abelian varieties of dimensions 1,2 and 3.

Higher-level canonical subgroups for p-divisible groups

AbstractLet R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti–Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗RK with geometric structure (Z/pnZ)g consisting of points ‘closest to zero’. We give a non-trivial condition on the Hasse invariant of G that guarantees the existence of the canonical subgroup, analogous to a result of Katz and Lubin for elliptic curves. The bound is independent of the height and dimension of G.

Canonical subgroups of Barsotti-Tate groups

Let S be the spectrum of a complete discrete valuation ring with fraction field of characteristic 0 and perfect residue field of characteristic p ≥ 3. Let be a truncated Barsotti-Tate group of level 1 over S. If G is not too supersingular, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, then we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fiber to a subgroup scheme of G, finite and flat over S. We call it the canonical subgroup of G.

$p$-adic monodromy of the universal deformation of a HW-cyclic Barsotti–Tate group

Let k be an algebraically closed field of characteristic p&gt;0 , and G be a Barsotti–Tate over k . We denote by \mathbf S the “algebraic” local moduli in characteristic p of G , by \mathbf G the universal deformation of G over \mathbf S , and by \mathbf U\subset\mathbf S the ordinary locus of \mathbf G . The étale part of \mathbf G over \mathbf U gives rise to a monodromy representation \rho_{\mathbf G} of the fundamental group of \mathbf U on the Tate module of \mathbf G . Motivated by a famous theorem of Igusa, we prove in this article that \rho_{\mathbf G} is surjective if G is connected and HW-cyclic. This latter condition is equivalent to saying that Oort's a -number of G equals 1, and it is satisfied by all connected one-dimensional Barsotti–Tate groups over k .

A compactification of Igusa varieties

We investigate the notion of Igusa level structure for a one-dimensional Barsotti–Tate group over a scheme X of positive characteristic and compare it to Drinfeld’s notion of level structure. In particular, we show how the geometry of the Igusa covers of X is useful for studying the geometry of its Drinfeld covers (e.g. connected and smooth components, singularities). Our results apply in particular to the study of the Shimura varieties considered in Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001). In this context, they are higher dimensional analogues of the classical work of Igusa for modular curves and of the work of Carayol for Shimura curves. In the case when the Barsotti–Tate group has constant p-rank, this approach was carried-out by Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001).

Potentially good reduction of Barsotti–Tate groups

Let R be a complete discrete valuation ring of mixed characteristic ( 0 , p ) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti–Tate group ( p-divisible group) defined over K which acquires good reduction over a finite extension K ′ of K. We prove that there exists a constant c ⩾ 2 which depends on the absolute ramification index e ( K ′ / Q p ) and the height of G such that G has good reduction over K if and only if G [ p c ] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “ p-adic Néron–Ogg–Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge–Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.

Monodromy of logarithmic Barsotti-Tate groups attached to 1-motives

Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of positive characteristic $p$ and field of fractions $K$ of characteristic 0. In this paper we consider a $K$-1-motive $M_K$ as in [Ra] and its associated Barsotti-Tate group. This last does not in general extend to a Barsotti-Tate group over $R$. However, with some assumptions, it extends to a logarithmic Barsotti-Tate group over $R$. This follows from [Ra] and Kato's results on finite logarithmic group schemes. Once chosen a uniformizing parameter $\pi$ of $R$, any logarithmic Barsotti-Tate group over $R$ is described by two data $(G,N)$ where $G$ is a classical Barsotti-Tate group over $R$ and $N$ is a homomorphism of classical Barsotti-Tate groups. Moreover, if $R=W(k)$, $N$ induces a $W(k)$-homorphism ${\cal N}\colon M(G_k)\to M(G_k)$ on Dieudonn\'e modules such that $F{\cal N}V={\cal N}$ and ${\cal N}^2=0$. In the first part of the paper we recall these constructions and we show how to relate $N$ with the ``geometric monodromy'' introduced by Raynaud. In the second part of the paper we give an explicit description of ${\cal N}$ in terms of additive extensions and integrals. In the last part of the paper we describe how to recover the logarithmic Barsotti-Tate group attached to a 1-motive from a concrete scheme endowed with a suitable logarithmic structure.

Fonction L unité d'un groupe de Barsotti-Tate

The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonne crystalline theory ([B-B-M]) to “crystal of level mG” in the sense of Berthelot. We show that the unit-root L-function of the Dieudonne crystal associated to G can be expressed in terms of the syntomic cohomology of the Ext group of G by the constant sheaf.