Bondarko M.V.
Finite flat commutative group schemes over complete discrete valuation rings III: classification, tangent spaces, and semistable reduction of Abelian varieties
Preprint series: Mathematica Gottingensis
MSC:
14L15 Group schemes
14L05 Formal groups, $p$-divisible groups, See also {55N22}
14G20 $p$-adic ground fields
11G10 Abelian varieties of dimension $\gtr 1$, See also {14Kxx}
11S31 Class field theory; $p$-adic formal groups, See also {14L05}
ZDM: F60
Abstract: The results from previous works are used to obtain a complete classification of finite local flat commutative group schemes over mixed characteristic complete discrete valuation rings in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes. In particular we prove that the minimal dimension of a formal group law that contains a given local group scheme $S$ as a closed subgroup is equal to the minimal number of generators for the affine algebra of $S$. As an application the following criterion of semistable reduction of Abelian varieties is proved.
Let $K$ be a mixed characteristic local field, let its residue field have characteristic $p$, $L$ be a finite extension of $K$, let $\mathfrak{O}_K\subset\mathfrak{O}_L$ be their rings of integers. Let $e$ be the absolute ramification index of $L$, $s=[\log_p(e/(p-1))]$, $e_0$ be the ramification index of $L/K$, $l=2s+v_p(e_0)+1$.
For a finite flat commutative $\mathfrak{O}_L$-group scheme $H$ we denote the $\mathfrak{O}_L$-dual of the module $J/J^2$ by $TH$. Here $J$ is the augmentation ideal of the affine algebra of $H$.
Let $V$ be an $m$-dimensional Abelian variety over $K$. Suppose that $V$ has semistable reduction over $L$.

Theorem.

$V$ has semistable reduction over $K$ if and only if
for some group scheme $H$ over $\mathfrak{O}_K$ there exist embeddings of $H_K$ into
$\operatorname{Ker}[p^{l}]_{V,K}$ and of $(\mathfrak{O}_L/p^l\mathfrak{O}_L)^m$ into $TH_\mathfrak{O}_L$.

We also prove a natural extension of Theorem A that gives an if and only if criterion for ordinary and good ordinary reduction.

Keywords: finite group scheme, Cartier module, tangent space, formal group, Abelian variety, semistable reduction, local field.