Semistable abelian varieties and maximal torsion 1-crystalline submodules
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 39-81.

Let p be a prime, let K be a discretely valued extension of 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/ 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 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 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/ 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 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1151
Classification: 11R33, 11R34, 14K15
Keywords: Néron component group, log 1-motive, torsion 1-crystalline representation
Gunton, Cody 1

1 Department of Mathematical Sciences Universitetsparken 5 University of Copenhagen 2100 Copenhagen Ø, Denmark
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Gunton, Cody. 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. http://archive.numdam.org/articles/10.5802/jtnb.1151/

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