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Adrian Vasiu

Publications and source records attributed to Adrian Vasiu.

48 records · Page 3Linked to original sources
arXiv2007

Level m stratifications of versal deformations of p-divisible groups

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d,m$ be positive integers. Let $D$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. Let $\scrD$ be a versal deformation of $D$ over a smooth $k$-scheme $\scrA$ which is equidimensional of dimension $cd$. We show that there exists a reduced, locally closed subscheme $\grs_D(m)$ of $\scrA$ that has the following property: a point $y\in\scrA(k)$ belongs to $\grs_D(m)(k)$ if and only if $y^*(\scrD)[p^m]$ is isomorphic to $D[p^m]$. We prove that $\grs_D(m)$ is {\it regular and equidimensional} of {\it dimension} $cd-\dim(\pmb{\text{Aut}}(D[p^m]))$. We give a proof of {\it Traverso's formula} which for $m>>0$ computes the codimension of $\grs_D(m)$ in $\scrA$ (i.e., $\dim(\pmb{\text{Aut}}(D[p^m]))$) in terms of the Newton polygon of $D$. We also provide a criterion of when $\grs_D(m)$ satisfies the {\it purity property} (i.e., it is an affine $\scrA$-scheme). Similar results are proved for {\it quasi Shimura $p$-varieties of Hodge type} that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic $(0,p)$.

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arXiv2007

Minimal truncations of supersingular p-divisible groups

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])).

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arXiv2007

Projective integral models of Shimura varieties of Hodge type with compact factors

Let $(G,X)$ be a Shimura pair of Hodge type such that $G$ is the Mumford--Tate group of some elements of $X$. We assume that for each simple factor $G_0$ of $G^{\ad}$ there exists a simple factor of $G_{0\dbR}$ which is compact. Let $N\Ge 3$. We show that for many compact open subgroups $K$ of $G(\dbA_f)$, the Shimura variety $\Sh(G,X)/K$ has a projective integral model $\scrN$ over $\dbZ[{1\over N}]$ which is a finite scheme over a certain Mumford moduli scheme $\scrA_{g,1,N}$. Equivalently, we show that if $A$ is an abelian variety over a number field and if the Mumford--Tate group of $A_{\dbC}$ is $G$, then $A$ has potentially good reduction everywhere. The last result represents significant progress towards the proof of a conjecture of Morita. If $\scrN$ is smooth over $\dbZ[{1\over N}]$, then it is a Néron model of its generic fibre. In this way one gets in arbitrary mixed characteristic, the very first examples of general nature of projective Néron models whose generic fibres are not finite schemes over abelian varieties.

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arXiv2007

Traverso's isogeny conjecture for p-divisible groups

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 $k$.

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arXiv2006

Crystalline boundedness principle

We prove that an $F$-crystal $(M,\vph)$ over an algebraically closed field $k$ of characteristic $p>0$ is determined by $(M,\vph)$ mod $p^n$, where $n\ge 1$ depends only on the rank of $M$ and on the greatest Hodge slope of $(M,\vph)$. We also extend this result to triples $(M,\vph,G)$, where $G$ is a flat, closed subgroup scheme of ${\bf GL}_M$ whose generic fibre is connected and has a Lie algebra normalized by $\vph$. We get two purity results. If ${\got C}$ is an $F$-crystal over a reduced ${\bf F}_p$-scheme $S$, then each stratum of the Newton polygon stratification of $S$ defined by ${\got C}$, is an affine $S$-scheme (a weaker result was known before for $S$ noetherian). The locally closed subscheme of the Mumford scheme ${\Ma_{d,1,N}}_k$ defined by the isomorphism class of a principally quasi-polarized $p$-divisible group over $k$ of height 2d, is an affine ${\Ma_{d,1,N}}_k$-scheme.

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arXiv2004

Surjectivity Criteria for p-adic Representations, Part II

We prove surjectivity criteria for $p$-adic representations and we apply them to abelian varieties over number fields. In particular, we provide examples of Jacobians over $\dbQ$ of dimension $d\in\{1,2,3\}$ whose 2-adic representations have as images $GSp_{2d}(\dbZ_2)$.

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arXiv2004

Surjectivity Criteria for p-adic Representations, Part I

We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism $G(W_2(k))\twoheadrightarrow G(k)$ has a section.

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arXiv2004

A purity theorem for abelian schemes

We prove a purity theorem for abelian schemes in arbitrary unramified mixed characteristic (0,p). The case p=2 is completely new and the case p>2 fixes several errors in the literature.

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arXiv2003

Good Reductions of Shimura Varieties of Preabelian Type in Arbitrary Unramified Mixed Characteristic, I

We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of integral canonical models of Shimura varieties of preabelian (resp. of abelian) type in mixed characteristic $(0,p)$ with $p\Ge 3$ (resp. with $p=2$) and with respect to hyperspecial subgroups; if $p=3$ (resp. if $p=2$) we restrict in this part I either to the $A_n$, $C_n$, $D_n^{\dbH}$ (resp. $A_n$ and $C_n$) types or to the $B_n$ and $D_n^{\dbR}$ (resp. $B_n$, $D_n^{\dbH}$ and $D_n^{\dbR}$) types which have compact factors over $\dbR$ (resp. which have compact factors over $\dbR$ in some $p$-compact sense). Though the second application is new just for $p\Le 3$, a great part of its proof is new even for $p\Ge 5$ and corrects [Va1, 6.4.11] in most of the cases. The second application forms progress towards the proof of a conjecture of Milne. It also provides in arbitrary mixed characteristic the very first examples of general nature of projective varieties over number fields which are not embeddable into abelian varieties and which have Néron models over certain local rings of rings of integers of number fields.

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arXiv2001

Shimura Varieties and the Mumford-Tate conjecture, part I

We prove the Mumford-Tate conjecture for those abelian varieties over number fields, whose simple factors of their adjoint Mumford-Tate groups have over $\dbR$ certain (products of) non-compact factors. In particular, we prove this conjecture for the orthogonal case ($B_n$ and $D_n^{\dbR}$ Shimura types). We construct embeddings of Shimura varieties of (whose adjoints are of prescribed abelian type) into unitary Shimura varieties. We also prove two general theorems involving Frobenius tori of abelian varieties over number fields.

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arXiv2001

Points of integral canonical models of Shimura varieties of preabelian type, p-divisible groups, and applications. First Part

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 generalized Shimura context, Dieudonné's theories (reobtained, simplified and extended), the main list of stratifications of special fibres of integral canonical models of Shimura varieties of preabelian type, the uniqueness of such models in mixed characteristic (0,2), the existence (in many situations) of such models in mixed characteristic (0,2), steps towards the classification of Shimura p-divisible groups over algebraically closed field of characteristic p (like the boundedness principle, the purity principle, Bruhat decompositions in the F-context, etc.), generalized Serre lemma, connections in Fontaine's categories, etc.

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