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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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