Semistable abelian varieties and maximal torsion 1-crystalline submodules

Gunton, Cody

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 39-81.

Soient $p$ un nombre premier, $K$ une extension de ${\mathbb{Q}}_p$ de valuation discrète, et $A_K$ une $K$-variété abélienne à réduction semistable. En étendant les travaux de Kim et Marshall portant sur le cas $p>2$ et $K/{\mathbb{Q}}_p$ non ramifié, nous prouvons un complément pour $l=p$ à la formule cohomologique de Grothendieck pour la partie $l$-primaire du groupe des composantes connexes du modèle de Néron de $A_K$. Notre démonstration consiste à construire, pour chaque $m\in {\mathbb{Z}}_{\ge 0}$, un schéma en groupes plat et fini sur ${𝒪}_K$ dont la fibre générique est isomorphe au sous-module $1$-cristallin maximal de $A_K\left[p^m\right]$. Comme corollaire, on obtient une nouvelle preuve du critère monodromique de bonne réduction de Coleman–Iovita.

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 ${𝒪}_K$-group scheme with generic fiber equal to the maximal $1$-crystalline submodule of $A_K\left[p^m\right]$. As a corollary, we have a new proof of the Coleman–Iovita monodromy criterion for good reduction of abelian $K$-varieties.

Classification : 11R33, 11R34, 14K15
Mots clés : Néron component group, log 1-motive, torsion 1-crystalline representation

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