On the cyclic torsion of elliptic curves over cubic number fields (II)

Wang, Jian¹

¹ College of Mathematics Jilin Normal University Siping, Jilin 136000, China

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 663-670.

Abstract (VO)

Merel’s result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell–Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=49,40,25$ or $22$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E{\left(K\right)}_{\mathrm{tor}}$ for any elliptic curve $E$ over a cubic number field $K$.

Résumé

Le résultat de Merel sur la forme forte de la conjecture de borne uniforme a mis en valeur la classification des parties de torsion des groupes de Mordell–Weil des courbes elliptiques définies sur les corps de nombres de degré fixé $d$. Dans cet article, nous étudions les sous-groupes de torsion cycliques des courbes elliptiques sur les corps de nombres cubiques. Pour $N=49,40,25$ ou $22$, nous montrons que $\mathbb{Z}/N\mathbb{Z}$ n’est pas un sous-groupe de $E{\left(K\right)}_{\mathrm{tor}}$ pour toute courbe elliptique $E$ sur un corps de nombres cubique $K$.

Détail

Reçu le : 2018-12-16
Révisé le : 2019-04-13
Accepté le : 2019-05-11
Publié le : 2020-05-06

DOI : 10.5802/jtnb.1100

Classification : 11G05, 11G18
Keywords: torsion subgroup, elliptic curves, modular curves

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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