We give a family of elliptic curves, depending on two nonzero rational parameters and , such that the following statement holds: let be an elliptic curve and let be its 3-torsion subgroup. This group verifies if and only if belongs to .
Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer to the local-global divisibility is unknown in the case of such algebraic groups. In this paper, we give a negative one. We show some curves of the family , with points locally divisible by 9 almost everywhere, but not globally, over a number field of degree at most 2 over .
Nous donnons une famille de courbes elliptiques, dépendant de deux paramètres rationnels non nuls et , telle que nous avons la propriété suivante : soit une courbe elliptique et soit son sous-groupe de 3-torsion. On a que si et seulement si est une courbe de la famille .
De plus, nous considérons le problème de la divisibilité locale-globale par 9 pour les points d’une courbe elliptique. Le nombre 9 est une des rares puissances d’un nombre premier pour laquelle on ne connait pas la réponse à la divisibilité locale-globale dans le cas de tels groupes algébriques. Dans ce papier nous donnons une réponse négative. Nous exhibons des courbes de la famille , avec des points qui sont localement divisibles par 9 presque partout, mais qui ne sont pas globalement divisibles par 9, sur un corps de nombres de degré au plus 2 sur .
@article{JTNB_2010__22_1_139_0, author = {Paladino, Laura}, title = {Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {139--160}, publisher = {Universit\'e Bordeaux 1}, volume = {22}, number = {1}, year = {2010}, doi = {10.5802/jtnb.708}, zbl = {1216.11064}, mrnumber = {2675877}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.708/} }
TY - JOUR AU - Paladino, Laura TI - Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9 JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 139 EP - 160 VL - 22 IS - 1 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.708/ DO - 10.5802/jtnb.708 LA - en ID - JTNB_2010__22_1_139_0 ER -
%0 Journal Article %A Paladino, Laura %T Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9 %J Journal de théorie des nombres de Bordeaux %D 2010 %P 139-160 %V 22 %N 1 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.708/ %R 10.5802/jtnb.708 %G en %F JTNB_2010__22_1_139_0
Paladino, Laura. Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 139-160. doi : 10.5802/jtnb.708. http://archive.numdam.org/articles/10.5802/jtnb.708/
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