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

Jian Wang

doi:10.5802/jtnb.1100

Jian Wang ¹

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.

Résumé
Abstract

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 $ℤ / N ℤ$ n’est pas un sous-groupe de $E {(K)}_{tor}$ pour toute courbe elliptique $E$ sur un corps de nombres cubique $K$ .

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 $ℤ / N ℤ$ is not a subgroup of $E {(K)}_{tor}$ for any elliptic curve $E$ over a cubic number field $K$ .

Reçu le : 2018-12-15
Révisé le : 2019-04-12
Accepté le : 2019-05-10
Publié le : 2020-05-06

MR Numdam Zbl

DOI : 10.5802/jtnb.1100

Classification : 11G05, 11G18
Mots clés : torsion subgroup, elliptic curves, modular curves

Affiliations des auteurs :

Jian Wang ¹

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the cyclic torsion of elliptic curves over cubic number fields {(II)}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {663--670},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Jian Wang. On the cyclic torsion of elliptic curves over cubic number fields (II). Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 663-670. doi : 10.5802/jtnb.1100. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1100/

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