Models of torsors under elliptic curves
Publications Mathématiques de Besançon (2017), pp. 79-108.

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.

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.

Publié le : 2017-11-30
DOI : https://doi.org/10.5802/pmb.16
Classification : 11G20,  14G05,  11G07
Mots clés: elliptic curves, torsors, curves of genus one, models, degenerations, dual graphs, rational points
@article{PMB_2017____79_0,
     author = {Kentaro Mitsui},
     title = {Models of torsors under elliptic curves},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2017},
     pages = {79-108},
     doi = {10.5802/pmb.16},
     mrnumber = {3752488},
     language = {en},
     url = {pmb.centre-mersenne.org/item/PMB_2017____79_0/}
}
Kentaro Mitsui. Models of torsors under elliptic curves. Publications Mathématiques de Besançon (2017), pp. 79-108. doi : 10.5802/pmb.16. https://pmb.centre-mersenne.org/item/PMB_2017____79_0/

[1] Vasyl Ī. Andriĭčuk The order and index of a principal homogeneous space of an elliptic curve over a general local field, Ukr. Mat. Zh., Volume 27 (1975), p. 62-63 | MR 371901 | Zbl 0338.14020

[2] Pete L. Clark The period-index problem in WC-groups IV: a local transition theorem, J. Théor. Nombres Bordx., Volume 22 (2010) no. 3, pp. 583-606 | Article | Numdam | MR 2769333 | Zbl 1258.11094

[3] Schémas en groupes I–III (Michel Demazure; Alexander Grothendieck, eds.), Lecture Notes in Mathematics, Volume 151, 152, 153, Springer, 1970, xv+564, ix+654, vii+529 pages (Séminaire de Géométrie Algébrique du Bois Marie 1962–1964 (SGA 3), Avec la collaboration de M. Artin, J.E. Bertin, P. Gabriel, M. Raynaud et J.-P. Serre) | Zbl 0212.52810

[4] Michael D. Fried; Moshe Jarden Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3., Volume 11, Springer, 2008, xxiv+792 pages (Revised by Jarden) | MR 2445111 | Zbl 1145.12001

[5] Ofer Gabber; Qing Liu; Dino Lorenzini The index of an algebraic variety, Invent. Math., Volume 192 (2013) no. 3, pp. 567-626 | Article | MR 3049930 | Zbl 1268.13009

[6] Silvio Greco Two theorems on excellent rings, Nagoya Math. J., Volume 60 (1976), pp. 139-149 | Article | MR 409452 | Zbl 0308.13010

[7] Alexander Grothendieck Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas (Seconde partie), Publ. Math., Inst. Hautes Étud. Sci., Volume 24 (1965), pp. 1-231 | Numdam | Zbl 0135.39701

[8] Alexander Grothendieck Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas (Quatrième partie), Publ. Math., Inst. Hautes Étud. Sci., Volume 32 (1967), pp. 1-361 | Numdam | Zbl 0153.22301

[9] Serge Lang Algebraic groups over finite fields, Am. J. Math., Volume 78 (1956), pp. 555-563 | Article | MR 86367 | Zbl 0073.37901

[10] Serge Lang; John Tate Principal homogeneous spaces over abelian varieties, Am. J. Math., Volume 80 (1958), pp. 659-684 | Article | MR 106226 | Zbl 0097.36203

[11] Stephen Lichtenbaum The period-index problem for elliptic curves, Am. J. Math., Volume 90 (1968), pp. 1209-1223 | Article | MR 237506 | Zbl 0187.18602

[12] Qing Liu Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, Volume 6, Oxford University Press, 2002, xv+576 pages | MR 1917232 | Zbl 0996.14005

[13] Qing Liu; Dino Lorenzini; Michel Raynaud Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math., Volume 157 (2004) no. 3, pp. 455-518 | Zbl 1060.14037

[14] Hideyuki Matsumura Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8, Cambridge University Press, 1989, xiv+320 pages (Translated from the Japanese by M. Reid) | MR 1011461 | Zbl 0666.13002

[15] James Stuart Milne Weil-Châtelet groups over local fields, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 273-284 | Article | Numdam | Zbl 0212.53201

[16] Jean-Pierre Serre Espaces fibrés algébriques (d’après André Weil), Séminaire Bourbaki, Vol. 2, Société Mathématique de France, 1995, p. 305-311 (Exp. No. 82)

[17] Jean-Pierre Serre Galois cohomology, Springer Monographs in Mathematics, Springer, Berlin, 2002, x+210 pages (Translated from the French by Patrick Ion and revised by the author) | Zbl 1004.12003

[18] Shahed Sharif Period and index of genus one curves over global fields, Math. Ann., Volume 354 (2012) no. 3, pp. 1029-1047 | Article | MR 2983078 | Zbl 1321.11064