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Torsion points on elliptic curves defined over quadratic fields. (English) Zbl 0647.14020
Let \(N\) be a positive integer composed of powers of \(2, 3, 5, 7, 11\) and 13, and denote by \(m\) a positive divisor of \(N\). It is proved that the open affine subscheme \(Y_ 1(m,N)\) of the modular curve \(X_ 1(m,N)\) has no rational points over any quadratic field \(k\), provided that \(X_ 1(m,N)\) is not hyperelliptic. This result leads to the conjecture that the only torsion groups which can occur for an elliptic curve \(E\) over a quadratic field \(k\) are of isomorphism type \(\mathbb Z/2\mathbb Z\times \mathbb Z/2n\mathbb Z\) for \(1\leq n\leq 6\), \(\mathbb Z/3n\mathbb Z\times \mathbb Z/3n\mathbb Z\) for \(1\leq n\leq 2\) over \(k=\mathbb Q(\sqrt{-3})\), \(\mathbb Z/4\mathbb Z\times \mathbb Z/4\mathbb Z\) over \(k=\mathbb Q(\sqrt{-1})\) or \(\mathbb Z/N\mathbb Z\) for \(1\leq N\leq 18\), \(N\neq 15, 17\).
The methods of proof are as follows. On the one hand, analogously to previous work of the authors, the non-existence of certain rational functions on a subcovering \(X_ 1(m,N)\to X\to X_ 0(N)\) is verified. On the other hand, the Mordell-Weil groups of the Jacobians on modular curves, in particular of the usual ones \(X_ 0(N)\) and \(X_ 1(N)\), are studied.
Reviewer’s remark: Probably due to a misprint, the group \(\mathbb Z/15\mathbb Z\) is missing in the author’s conjecture. We mention in this connection that M. A. Reichert in Math. Comput. 46, 637–658 (1986; Zbl 0605.14028) has constructed elliptic curves \(E\) over quadratic fields \(k\) having torsion groups \(E_{tor}(k)\cong \mathbb Z/N\mathbb Z\) for \(11\leq N\leq 18\), \(N\neq 17\).
Reviewer: H. G. Zimmer

MSC:
11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
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