Elliptic curves with ([3])=(ζ 3 ) and counterexamples to local-global divisibility by 9
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 139-160.

Nous donnons une famille h,β de courbes elliptiques, dépendant de deux paramètres rationnels non nuls β et h, telle que nous avons la propriété suivante : soit une courbe elliptique et soit [3] son sous-groupe de 3-torsion. On a que ([3])=(ζ 3 ) si et seulement si est une courbe de la famille h,β .

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 h,β , 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 (ζ 3 ).

We give a family h,β of elliptic curves, depending on two nonzero rational parameters β and h, such that the following statement holds: let be an elliptic curve and let [3] be its 3-torsion subgroup. This group verifies ([3])=(ζ 3 ) if and only if belongs to h,β .

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 h,β , with points locally divisible by 9 almost everywhere, but not globally, over a number field of degree at most 2 over (ζ 3 ).

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DOI : https://doi.org/10.5802/jtnb.708
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     title = {Elliptic curves with ${\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3)$ and counterexamples to local-global divisibility by 9},
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Laura Paladino. 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, Tome 22 (2010) no. 1, pp. 139-160. doi : 10.5802/jtnb.708. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.708/

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