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Rational points on elliptic curves over \({\mathbb{Q}}\) in elementary abelian 2-extensions of \({\mathbb{Q}}\). (English) Zbl 0586.14013
Let E be an elliptic curve over \({\mathbb{Q}}\) and let F be the maximal abelian 2-extension of \({\mathbb{Q}}\). The authors give an explicit list of 22 finite abelian groups (of order dividing \(2^ 7\cdot 3^ 2\cdot 5\cdot 7)\) and they show that the torsion subgroup of E(F) is isomorphic to one of the groups in their list. They prove this by first remarking that it suffices to show that for finite \({\mathbb{Q}}\)-extensions \(K={\mathbb{Q}}(\sqrt{d_ 1},...,\sqrt{d_ n})\subset F\) no other groups can occur than certain subgroups of the ones in the list. This is done by comparing E(K) to \(\oplus E^{(d_ i)}({\mathbb{Q}}),\quad where\) \(E^{(d)}\) denotes the quadratic twist of E/\({\mathbb{Q}}\) over the field \({\mathbb{Q}}(\sqrt{d})\). The torsion groups of the groups \(E^{(d_ i)}({\mathbb{Q}})\) are well-known by the work of B. Mazur [Publ. Math., Inst. Hautes √Čtud. Sci. 47(1977), 33-186 (1978; Zbl 0394.14008); p. 145]. Various papers of M. A. Kenku on rational points on certain modular curves [J. Lond. Math. Soc., II. Ser. 11, 93-98 (1975; Zbl 0313.14002); 19, 233-240 (1979; Zbl 0431.14007); 22, 239-244 (1980; Zbl 0437.14022), and 23, 415-427 (1981; Zbl 0425.14006) and 23, 428 (1981; Zbl 0468.14012); Math. Proc. Camb. Philos. Soc. 85, 21-23 (1979; Zbl 0392.14011); 87, 15-20 (1980; Zbl 0479.14014)] are used to finish the proof.
Reviewer: J.Top

MSC:
14G05 Rational points
14H52 Elliptic curves
14H25 Arithmetic ground fields for curves
14H45 Special algebraic curves and curves of low genus
11R11 Quadratic extensions
14G25 Global ground fields in algebraic geometry
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