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On Mazur’s conjecture. (English. Russian original) Zbl 0944.14012
Math. Notes 63, No. 2, 294-296 (1998); translation from Mat. Zametki 63, No. 2, 255-257 (1998).
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.
MSC:
14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
11R20 Other abelian and metabelian extensions
11G05 Elliptic curves over global fields
14G05 Rational points
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[2] V. A. Dem’yanenko,Mat. Zametki [Math. Notes],37, No. 1, 99–102 (1985).
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