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

Wang, Jian

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

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 $K$.

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 $K$.

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

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