On rational torsion points of central $\mathbb{Q}$-curves

Sairaiji, Fumio; Yamauchi, Takuya

Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 465-483.

Let $E$ be a central $\mathbb{Q}$-curve over a polyquadratic field $k$. In this article we give an upper bound for prime divisors of the order of the $k$-rational torsion subgroup $E_{tors}\left(k\right)$ (see Theorems 1.1 and 1.2). The notion of central $\mathbb{Q}$-curves is a generalization of that of elliptic curves over $\mathbb{Q}$. Our result is a generalization of Theorem 2 of Mazur [12], and it is a precision of the upper bounds of Merel [15] and Oesterlé [17].

Soit $E$ une $\mathbb{Q}$-courbe centrale sur un corps polyquadratique $k$. Dans cet article, nous donnons une borne supérieure des diviseurs premiers de l’ordre du sous-groupe de torsion $k$-rationnel $E_{tors}\left(k\right)$ (voir Théorèmes 1.1 et 1.2). La notion de $\mathbb{Q}$-courbe centrale est une généralisation de celle de courbe elliptique sur $\mathbb{Q}$. Notre résultat est une généralisation du Théorème de Mazur [12], et c’est une précision des bornes supérieures de Merel [15] et Oesterlé [17].

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