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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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