Semistable abelian varieties and maximal torsion 1-crystalline submodules

TL;DR

This paper extends Grothendieck's Galois cohomological formula to the $l=p$ case for the Néron component group of semistable abelian varieties over local fields, using new constructions of finite flat group schemes.

Contribution

It generalizes previous results to the case where $p$ divides the prime, providing a new proof of the Coleman-Iovita monodromy criterion through novel group scheme constructions.

Findings

Extended Grothendieck's formula to $l=p$ case.

Constructed finite flat group schemes with specific properties.

Provided a new proof of the Coleman-Iovita criterion.

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 N\'eron component group of $A_{K}$. Our proof involves constructing, for each $m\in \mathbb{Z}_{\geq 0}$, a finite flat $\mathscr{O}_K$-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.