Finite flat commutative group schemes over complete discrete valuation fields: classification, structural results; application to reduction of Abelian varieties

Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_{\eta}$. Let $f: Y \rightarrow X_{\eta}$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_{\eta}$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue char...

We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalising a classical result of Raynaud in two different ways. The first is an extension of Raynaud's numerical criterion to higher dimensions. The second is a logarithmic criterion: we show that a proper, fs log smooth morphism to a log regular discrete valuation ring is cohomologically flat in dimension 0. We apply this latter result to curves and torsors under abelia...

This article is superseded by 1703.06631. We keep this version here since some of the arguments for the special cases treated here are different than those of 1703.06631.

Let $K$ be a complete discrete valuation field. Let $\mathcal{O}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\geq1$. Let $G/K$ be a semi-abelian variety. Let $\mathcal{G}/\mathcal{O}_K$ be its N\'eron model. The special fiber $\mathcal{G}_k/k$ is an extension of the identity component $\mathcal{G}_k^0/k$ by the group of components $\Phi(G)$. We say that $G/K$ has split reduction if this exten...

Let $R$ be a discrete valuation ring of field of fractions $K$ and of residue field $k$ of characteristic $p > 0$.\\ In an earlier work, we studied the question of extending torsors on $K$-curves into torsors over $R$-regular models of the curves in the case when the structural $K$-group scheme of the torsor admits a finite flat model over $R$. In this paper, we first give a simpler description of the problem in the case where the curve is semistable. Secondly, if $R$ is assu...

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular implies Grothendieck's conjecture on the perfectness of his pairing between the Neron component groups of an abelian variety and its dual. The point is that our formulation is well-suited with Galois descent. From the known case of semistable abelian varieties, we...

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give necessary and sufficient conditions for the Hodge group to be semisimple. We obtain bounds on certain torsion subgroups for abelian varieties which do not have purely multiplicative reduction at a given discrete valuation, and therefore obtai...

For a prime number p>2, we give a direct proof of Breuil's classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings's strict modules can be identified via the Fontaine-Wintenberger field-of-norms functor.

Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction of $\mathcal{A}$ at the prime ideal. In this paper we derive an algorithm which allows to decompose the group scheme ${\rm A}[p]$ into indecomposable quasi-polarized ${\rm BT}_1$-group schemes. This can be done for the unramified $p$ on the ...

The Deligne-Mumford stable reduction theorem asserts that for a family of stable curves over the punctured disk, after a finite base change, the family can be completed in a unique way to a family of stable curves over the disk. In this survey we discuss stable reduction theorems in a number of different contexts. This includes a review of recent results on abelian varieties, canonically polarized varieties, and singularities. We also consider the semi-stable reduction theore...