Let p be a prime, let K be a discretely valued extension of
Qp, and let AK be an abelian K-variety with semistable
reduction. Extending work by Kim and Marshall from the case where p>2 and
K/Qp is unramified, we prove an l=p complement of a Galois
cohomological formula of Grothendieck for the l-primary part of the N\'eron
component group of AK. Our proof involves constructing, for each m∈Z≥0, a finite flat OK-group scheme with generic
fiber equal to the maximal 1-crystalline submodule of AK[pm]. As a
corollary, we have a new proof of the Coleman-Iovita monodromy criterion for
good reduction of abelian K-varieties.Comment: Fixed typos and added funding acknowledgemen