Mathematics > Number Theory
[Submitted on 19 Dec 2018 (v1), last revised 29 Aug 2021 (this version, v3)]
Title:Semistable abelian varieties and maximal torsion 1-crystalline submodules
View PDFAbstract: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 $m\in \mathbb{Z}_{\geq 0}$, 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.
Submission history
From: Cody Gunton [view email][v1] Wed, 19 Dec 2018 13:38:25 UTC (43 KB)
[v2] Tue, 14 Jan 2020 11:39:42 UTC (43 KB)
[v3] Sun, 29 Aug 2021 19:11:42 UTC (48 KB)
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