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Elliptic functions and the Fermat curve. (English. Russian original) Zbl 0915.14016
Math. Notes 60, No. 4, 453-456 (1996); translation from Mat. Zametki 60, No. 4, 606-608 (1996).
In this paper the following situation is considered: Let $$k$$ be an algebraic number field, and $${\mathcal F}$$ the curve $$y^2=x^3+rx+s$$, where $$4r^3 + 27s^2\neq 0$$, while $$\{ {\mathcal C}_{m,1}, {\mathcal C}_{m,2} \}$$ is a basis of all points of order $$m$$ on $${\mathcal F}$$. When $${\mathcal C}_{m,1}\in k$$ and $$m=2^5$$, $$r=1$$, $$s=0$$, the points of $${\mathcal F}$$ on $$k$$ are described; while when $$m=3^3$$ the points of the curve $$x^3+y^3=1$$ over $$k(\sqrt{-3})$$ are described. Then the following conjecture (in the line of Mazur conjecture) is expressed:
If the order of torsion of $${\mathcal F}$$ over $$k$$ is $$m$$, then the order of torsion of it over $$k(e^{{2\pi i \over m}})$$ is also $$m$$.
Some cases are discussed.
##### MSC:
 14H25 Arithmetic ground fields for curves 14G05 Rational points 14G25 Global ground fields in algebraic geometry 11R11 Quadratic extensions
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##### References:
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