Let p be a prime, let K be a discretely valued extension of Q_p, and let A_K be an abelian K-variety with semistable reduction. Extending work by Kim and Marshall from the case where p > 2 and K/Q_p is unramified, we prove an l = p complement of a Galois cohomological formula of Grothendieck for the l-primary part of the Néron component group of A_K . Our proof in-volves constructing, for each m ∈ Z_≥0, a finite flat O_K-group scheme with generic fiber equal to the maximal 1-crystalline submodule of A_K [p^m ]. As a corollary, we have a new proof of the Coleman–Iovita monodromy criterion for good reduction of abelian K-varieties.