Semistable abelian varieties and maximal torsion 1-crystalline submodules

Cody Gunton

doi:10.5802/jtnb.1151

Cody Gunton ¹

¹ Department of Mathematical Sciences Universitetsparken 5 University of Copenhagen 2100 Copenhagen Ø, Denmark

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

Abstract
Résumé

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 $m \in ℤ_{\geq 0}$ , 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.

Soient $p$ un nombre premier, $K$ une extension de $ℚ_{p}$ 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 $K / ℚ_{p}$ 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 $m \in ℤ_{\geq 0}$ , un schéma en groupes plat et fini sur $𝒪_{K}$ dont la fibre générique est isomorphe au sous-module $1$ -cristallin maximal de $A_{K} [p^{m}]$ . Comme corollaire, on obtient une nouvelle preuve du critère monodromique de bonne réduction de Coleman–Iovita.

Received: 2020-01-15
Revised: 2020-08-27
Accepted: 2021-02-01
Published online: 2021-05-21

DOI: 10.5802/jtnb.1151

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

Author's affiliations:

Cody Gunton ¹

¹ Department of Mathematical Sciences Universitetsparken 5 University of Copenhagen 2100 Copenhagen Ø, Denmark

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Semistable abelian varieties and maximal torsion 1-crystalline submodules},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {39--81},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Cody Gunton. Semistable abelian varieties and maximal torsion 1-crystalline submodules. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 39-81. doi : 10.5802/jtnb.1151. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1151/

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