Semistable abelian varieties and maximal torsion 1-crystalline submodules

Abstract

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 semistable reduction. 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 Galois cohomological formula of Grothendieck for the $l$-primary part of the Néron component group of $A_K$. Our proof involves constructing, for each $m\in {\mathbb{Z}}_{\ge 0}$, a finite flat ${\mathcal{O}}_K$-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.

Final version

References in corpus (1)

• Crystalline realizations of 1-motives