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Let $p$ be a prime, let $K$ be a discretely valued extension of$\mathbb{Q}_p$, and let $A_{K}$ be an abelian $K$-variety with semistablereduction. Extending work by Kim and Marshall from the case where $p>2$ and$K/\mathbb{Q}_p$ is unramified, we prove an $l=p$ complement of a Galoiscohomological formula of Grothendieck for the $l$-primary part of the N\'eroncomponent group of $A_{K}$. Our proof involves constructing, for each $m\in\mathbb{Z}_{\geq 0}$, a finite flat $\mathscr{O}_K$-group scheme with genericfiber equal to the maximal 1-crystalline submodule of $A_{K}[p^{m}]$. As acorollary, we have a new proof of the Coleman-Iovita monodromy criterion forgood reduction of abelian $K$-varieties.

Semistable abelian varieties and maximal torsion 1-crystalline submodules