Models of torsors under elliptic curves
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2017), pp. 79-108.

We study the special fibers of the minimal proper regular models of proper smooth geometrically integral curves of genus one over a complete discrete valuation field. We classify the configurations of their irreducible components when the residue field is perfect. As an application, we show the existence of separable closed points of small degree on the original curves when the residue field is finite. Finally, we extend this result under mild assumptions on the residue field and the degenerations of their Jacobians.

Nous étudions les fibres spéciales des modèles propres réguliers minimaux de courbes propres lisses géométriquement intègres de genre un sur un corps de valuation discrète complet. Nous classifions les configurations de leurs composantes irréductibles quand le corps résiduel est parfait. En guise d’application, nous montrons l’existence de points fermés séparables de petit degré des courbes originales quand le corps résiduel est fini. Finalement, nous étendons ce résultat sous des hypothèses faibles sur le corps résiduel et la dégénérescence de la jacobienne.

Published online:
DOI: 10.5802/pmb.16
Classification: 11G20,  14G05,  11G07
Keywords: elliptic curves, torsors, curves of genus one, models, degenerations, dual graphs, rational points
Mitsui, Kentaro 1

1 Department of Mathematics, Graduate School of Science, Kobe University, Hyogo 657-8501, Japan
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Mitsui, Kentaro. Models of torsors under elliptic curves. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2017), pp. 79-108. doi : 10.5802/pmb.16. http://archive.numdam.org/articles/10.5802/pmb.16/

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