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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
##### Keywords:
Mazur’s conjecture; elliptic curves; algebraic fields
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##### References:
 [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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