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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.
14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
11R20 Other abelian and metabelian extensions
11G05 Elliptic curves over global fields
14G05 Rational points
Full Text: DOI
[1] M. I. Bashmakov and A. S. Kirillov,Zap. Nauchn. Sem. LOMI,57, 5–7 (1976).
[2] V. A. Dem’yanenko,Mat. Zametki [Math. Notes],37, No. 1, 99–102 (1985).
[3] V. A. Dem’yanenko,Zap. Nauchn. Sem. LOMI,111, 58–61 (1983).
[4] M. I. Bashmakov and al’-Nader,Mat. Sb. [Math. USSR-Sb],90, No. 1, 117–130 (1973). · Zbl 0273.14010 · doi:10.1070/SM1973v019n01ABEH001739
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