**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}

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.