Barsotti-Tate Groups Research Articles

Rank 2 local systems, Barsotti–Tate groups, and Shimura curves

We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.

Equivariant formal group laws and complex-oriented spectra over primary cyclic groups: elliptic curves, Barsotti–Tate groups, and other examples

We explicitly construct and investigate a number of examples of $${\mathbb {Z}}/p^r$$ -equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and p-divisible groups, as well as some other related examples.

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 The Hecke Orbit Conjecture for PEL Type Shimura Varieties

The Hecke orbit conjecture plays an important role in understanding the geometric structure of Shimura varieties. First postulated by Chai and Oort in 1995, the Hecke orbit conjecture predicts that prime-to-p Hecke correspondences on mod p reductions of Shimura varieties characterize the foliation structure formed by Oort's central leaves. In other words, every prime-to-p Hecke orbit is Zariski dense in the central leaf containing it. Roughly speaking, a central leaf is the locus in a Shimura variety consisting of all points whose corresponding Barsotti-Tate groups belong to a fixed geometric isomorphism class. On the other hand, the prime-to-p Hecke orbit of a closed point x is the (countable) set consisting of all points y such that there is a prime-to-p quasi-isogeny from x to y. In 2005, Chai and Yu proved the Hecke orbit conjecture for Hilbert modular varieties, followed by a proof for Siegel modular varieties by Chai and Oort in the same year. The major purpose of the present work is to generalize the method of Chai and Oort to Shimura varieties of PEL type. We show that the Hecke orbit conjecture holds for points in certain irreducible components of Newton strata under our assumptions.

On the Chow ring of the stack of truncated Barsotti–Tate groups

We determine the Chow ring of the stack of truncated displays and more generally the Chow ring of the stack of G-zips. We also investigate the pull-back morphism of the truncated display functor. From this we can determine the Chow ring of the stack of truncated Barsotti-Tate groups over a field of characteristic p up to p-torsion.

The zeta functions of moduli stacks of G-zips and moduli stacks of truncated Barsotti-Tate groups

We study stacks of truncated Barsotti-Tate groups and the G-zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height 1 with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group G and its action on the root system. The main ingredients are the classification of G-zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, and the study of truncated Barsotti-Tate groups and their automorphism groups by Gabber & Vasiu.

The zeta Functions of Moduli Stacks of $G$-Zips and Moduli Stacks of Truncated Barsotti-Tate Groups

We study stacks of truncated Barsotti-Tate groups and the G -zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height 1 with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group G and its action on the root system. The main ingredients are the classification of G -zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, and the study of truncated Barsotti-Tate groups and their automorphism groups by Gabber & Vasiu.

Breuil 𝒪-windows and 𝜋-divisible 𝒪-modules

Let p > 2 p>2 be a prime number. Let O \mathcal {O} be the ring of integers of a finite extension of Q p \mathbb {Q}_p and let π \pi be a uniformizer of O \mathcal {O} . We prove that, for any complete Noetherian regular local O \mathcal {O} -algebra R R with perfect residue field of characteristic p p , the category of Breuil O \mathcal {O} -windows over R R is equivalent to the category of π \pi -divisible O \mathcal {O} -modules over R R . We also prove that the category of Breuil O \mathcal {O} -modules over R R is equivalent to the category of commutative finite flat O \mathcal {O} -group schemes over R R which are kernels of isogenies of π \pi -divisible O \mathcal {O} -modules. As an application of these equivalences, we then prove a boundedness result on Barsotti-Tate groups and deduce some corollaries. The results generalize some earlier results of Zink, Vasiu-Zink, and Lau.

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.

TRUNCATED BARSOTTI–TATE GROUPS AND DISPLAYS

We define truncated displays over rings in which a prime $p$ is nilpotent, we associate crystals to truncated displays, and we define functors from truncated displays to truncated Barsotti–Tate groups.

On $F$-crystalline representations

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension F/\mathbb Q_p , and an arbitrary finite extension K/F , we construct a general class of infinite and totally wildly ramified extensions K_\infty/K so that the functor V\mapsto V|_{G_{K_\infty}} is fully-faithfull on the category of F -crystalline representations V . We also establish a new classification of F -Barsotti–Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.

F-zips with additional structure

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G$. The other incarnation is a certain $G$-torsor analogue of the notion of $F$-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form $[E_{G,\chi}\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others. For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary $F$-zips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) and to truncated Barsotti-Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.

The correspondence between Barsotti-Tate groups and Kisin modules when p=2

Let K be a finite extension over Q2 and OK the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over OK .

Semi-Abelian Schemes and Heights of Cycles in Moduli Spaces of Abelian Varieties

We study extension properties of Barsotti-Tate groups and we establish diophantine inequalities involving heights of cycles with respect to logarithmically singular hermitian line bundles on arithmetic varieties. We apply these results to bound heights of cycles on moduli spaces of abelian varieties, induced by quotients of abelian varieties by levels of Barsotti-Tate subgroups, over function fields over number fields. To achieve this aim, we combine our results with an effective version of Rumely's theorem on integral points on possibly open arithmetic surfaces and with Faltings' theorems on heights of abelian varieties in isogeny classes.

Dimensions of Group Schemes of Automorphisms of Truncated Barsotti-Tate Groups

Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $\gamma_D(i):=\dim(\pmb{Aut}(D[p^i]))$. We show that $n_D\le cd$, $(\gamma_D(i+1)-\gamma_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb N$, for $cd>0$ we have $\gamma_D(1)<\gamma_D(2)<...<\gamma_D(n_D)$, and for $m\in\{1,...,n_D-1\}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.

Smoothness of the truncated display functor

We show that to every p p -divisible group over a p p -adic ring one can associate a display by crystalline Dieudonné theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal p p -divisible groups and nilpotent displays over p p -adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of p p -power order and Dieudonné modules over perfect rings.

Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristic

Let p be a prime. Let V be a discrete valuation ring of mixed characteristic (0,p) and index of ramification e. Let f:G→H be a homomorphism of finite flat commutative group schemes of p power order over V whose generic fiber is an isomorphism. We provide a new proof of a result of Bondarko and Liu that bounds the kernel and the cokernel of the special fiber of f in terms of e. For e<p−1 this reproves a result of Raynaud. Our bounds are sharper that the ones of Liu, are almost as sharp as the ones of Bondarko, and involve a very simple and short method. As an application we obtain a new proof of an extension theorem for homomorphisms of truncated Barsotti–Tate groups which strengthens Tateʼs extension theorem for homomorphisms of p-divisible groups.

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.

Successive extensions of minimal [formula omitted]-divisible groups

We study truncated Barsotti–Tate groups of level one ( BT 1 ) and their extensions to p -divisible groups. Firstly we show that any BT 1 contains a certain minimal BT 1 as a non-zero subgroup scheme. This proves that any BT 1 is written as a successive extension of minimal BT 1 ’s. Secondly we prove that any successive extension of minimal BT 1 ’s which is a BT 1 can be extended to a certain successive extension of minimal p -divisible groups. As an application, we determine the optimal upper bound of the last Newton slopes of p -divisible groups with a given isomorphism type of p -kernel.

$p$-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 .