Semistable abelian varieties and maximal torsion 1-crystalline submodules

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