Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
With cedram.org version française
Table of contents for this volume | Previous article | Next article
Kentaro Mitsui
Models of torsors under elliptic curves
Publications mathématiques de Besançon (2017), p. 79-108, doi: 10.5802/pmb.16
Article PDF
Class. Math.: 11G20, 14G05, 11G07
Keywords: elliptic curves, torsors, curves of genus one, models, degenerations, dual graphs, rational points

Abstract

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.

Bibliography

[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. 27 (1975), p. 62-63
[2] Pete L. Clark, “The period-index problem in WC-groups IV: a local transition theorem”, J. Théor. Nombres Bordx. 22 (2010) no. 3, p. 583-606
[3] Michel Demazure & Alexander Grothendieck (ed.), Schémas en groupes I–III, Lecture Notes in Mathematics 151, 152, 153, Springer, 1970, 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
[4] Michael D. Fried & Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. 11, Springer, 2008, Revised by Jarden
[5] Ofer Gabber, Qing Liu & Dino Lorenzini, “The index of an algebraic variety”, Invent. Math. 192 (2013) no. 3, p. 567-626
[6] Silvio Greco, “Two theorems on excellent rings”, Nagoya Math. J. 60 (1976), p. 139-149
[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. 24 (1965), p. 1-231
[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. 32 (1967), p. 1-361
[9] Serge Lang, “Algebraic groups over finite fields”, Am. J. Math. 78 (1956), p. 555-563
[10] Serge Lang & John Tate, “Principal homogeneous spaces over abelian varieties”, Am. J. Math. 80 (1958), p. 659-684
[11] Stephen Lichtenbaum, “The period-index problem for elliptic curves”, Am. J. Math. 90 (1968), p. 1209-1223
[12] Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press, 2002
[13] Qing Liu, Dino Lorenzini & Michel Raynaud, “Néron models, Lie algebras, and reduction of curves of genus one”, Invent. Math. 157 (2004) no. 3, p. 455-518
[14] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1989, Translated from the Japanese by M. Reid
[15] James Stuart Milne, “Weil-Châtelet groups over local fields”, Ann. Sci. Éc. Norm. Supér. 3 (1970), p. 273-284
[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, Translated from the French by Patrick Ion and revised by the author
[18] Shahed Sharif, “Period and index of genus one curves over global fields”, Math. Ann. 354 (2012) no. 3, p. 1029-1047

UFC LMB PUFC