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Détermination explicité des courbes elliptiques ayant un groupe de torsion non trivial sur des corps de nombres quadratiques sur \(\mathbb{Q}\). (French) Zbl 0562.14009
Sémin. Théor. Nombres, Univ. Bordeaux I 1983-1984, Exp. No. 11, 33 p. (1984).
This seminar talk is a report on the author’s diploma theses [the author, ”Explizite Bestimmung nichttrivialer Torsionsstrukturen elliptischer Kurven über quadratischen Zahlkörpern” (Saarbrücken 1983)]. By Mazur’s theorem the torsion subgroup of the group of \({\mathbb{Q}}\)-rational points of an elliptic curve E defined over \({\mathbb{Q}}\) is either cyclic of order \(m=1,2,3,...,10\) or 12 or isomorphic to \({\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2m{\mathbb{Z}}\) for \(m=1,2,3,4\). The paper under review deals with the problem of finding elliptic curves which admit a bigger (cyclic) torsion subgroup at the cost of allowing a field of rationality (both for E and the torsion points) which is a quadratic extension K/\({\mathbb{Q}}\). After the determination of a defining equation of the modular curves \(X_ 1(N)\) for \(N=11, 13, 14, 15, 16, 17, 18\) it is indicated how to show that for \(K={\mathbb{Q}}(\sqrt{-1})\) or \({\mathbb{Q}}(\sqrt{-11})\) there is no elliptic curve defined over K having a K-rational point of order 11. Based on a method after Billing, Mahler and Kubert [G. Billing and K. Mahler, J. Lond. Math. Soc. 15, 32-43 (1940; Zbl 0026.19901); D. S. Kubert, Proc. Lond. Math. Soc., III. Ser. 33, 193-237 (1976; Zbl 0331.14010)] the author uses his explicit equations for \(X_ 1(N)\) to compute a table of examples of elliptic curves E defined over certain quadratic fields K/\({\mathbb{Q}}\), such that the group of K-rational points of E has the torsion subgroup \(E_{tor}(K)={\mathbb{Z}}/m{\mathbb{Z}}\) for \(m=11, 13, 14, 15, 16\) and 18.
Reviewer: C.-G.Schmidt

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