On Mazur's conjecture (Q1272290)

Let \(n\) be a positive integer, \(\varepsilon (n) = e^{2\pi i\over n}\), \(F\) an elliptic curve over \({\mathbb Q}(\varepsilon (n))\), and \(\{O_m ,O_m'\}\) the basis of all points of order \(m\) on \(F\); in this paper it is proved that (1) if \(O_{2^\alpha}(F)\), \(2^{\alpha -1}O_{2^\alpha}'(F)=O _{2}'(F)\in {\mathbb Q}(\varepsilon (2^t))\), where \(t\) is arbitrary, then \(\alpha \leq 3\); (2) if \(O_{3^\beta}(F)\),\(O_{3^\beta}'(F)\in {\mathbb Q}(\varepsilon (3^t))\), where \(t\) is arbitrary, then \(\beta \leq 1\). The result yields to a conjecture about the groups of \(m\)-torsion points on two elliptic curves over an algebraic field.