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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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