Semistable reduction and torsion subgroups of abelian varieties

Silverberg, Alice; Zarhin, Yuri G.

doi:10.5802/aif.1459

Silverberg, Alice ; Zarhin, Yuri G.

Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 403-420.

Résumé
Abstract

Le résultat principal de cet article implique que si une variété abélienne définie sur un corps $F$ a un sous-groupe isotropique maximal dont les points d’ordre $n$ sont définis sur $F$ , et $n \geq 5$ , alors la variété abélienne a une réduction semi-stable en dehors de $n$ . Ce résultat peut être considéré comme une extension du théorème de Raynaud que si une variété abélienne a tous ses points d’ordre $n$ définis sur $F$ avec $n \geq 3$ , alors la variété abélienne a une réduction semi-stable en dehors de $n$ . Nous donnons aussi des renseignements sur les modèles de Néron dans les cas où $n = 2, 3$ ou 4.

The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$ -torsion points all of which are defined over $F$ , and $n \geq 5$ , then the abelian variety has semistable reduction away from $n$ . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$ -torsion points are defined over a field $F$ and $n \geq 3$ , then the abelian variety has semistable reduction away from $n$ . We also give information about the Néron models in the cases where $n = 2, 3$ and 4.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1459

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     title = {Semistable reduction and torsion subgroups of abelian varieties},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {2},
     year = {1995},
     doi = {10.5802/aif.1459},
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     zbl = {0818.14017},
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Silverberg, Alice; Zarhin, Yuri G. Semistable reduction and torsion subgroups of abelian varieties. Annales de l'Institut Fourier, Tome 45 (1995) no. 2, pp. 403-420. doi : 10.5802/aif.1459. https://www.numdam.org/articles/10.5802/aif.1459/

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