Advanced Search

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 [], and it is a precision of the upper bounds of Merel [] and Oesterlé [].