Semistable abelian varieties and maximal torsion 1-crystalline submodules

Abstract

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 involves 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.