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Miyake, Katsuya
Twists of hessian elliptic curves and cubic fields. Annales mathématiques Blaise Pascal, 16 no. 1 (2009), p. 27-45
Texte intégral djvu | pdf | Analyses MR 2514525 | Zbl 1182.11026
Class. Math.: 11G05, 12F05

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In this paper we investigate Hesse’s elliptic curves $H_{\mu } : U^3 + V^3 + W^3 = 3\mu UVW, \mu \in \mathbf{Q} - \lbrace 1\rbrace $, and construct their twists, $H_{\mu , t}$ over quadratic fields, and $\tilde{H}(\mu , t), \mu , t \in \mathbf{Q}$ over the Galois closures of cubic fields. We also show that $H_{\mu }$ is a twist of $\tilde{H}(\mu , t)$ over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, $R(t; X) := X^3 + tX + t, t \in \mathbf{Q} - \lbrace 0, - 27/4 \rbrace $, to parametrize all of quadratic fields and cubic ones. It should be noted that $\tilde{H}(\mu , t)$ is a twist of $H_{\mu }$ as algebraic curves because it may not always have any rational points over $\mathbf{Q}$. We also describe the set of $\mathbf{Q}$-rational points of $\tilde{H}(\mu , t)$ by a certain subset of the cubic field. In the case of $\mu = 0$, we give a criterion for $\tilde{H}(0, t)$ to have a rational point over $\mathbf{Q}$.


[1] Akinari Hoshi and Katsuya Miyake, Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation, Proc. Japan Acad. Ser. A Math. Sci., 833:21-26, 2007
Article |  MR 2317305 |  Zbl 1126.14018
[2] Dale Husemoller, Elliptic curves, Graduate Texts in Mathematics 111, Springer-Verlag, 1987  MR 868861 |  Zbl 0605.14032
[3] Katsuya Miyake, Some families of Mordell curves associated to cubic fields, Proceedings of the International Conference on Special Functions and their Applications (Chennai, 2002), 160, pages 217-231, 2003  MR 2022613 |  Zbl 1080.14520
[4] Katsuya Miyake, An introduction to elliptic curves and their Diophantine geometry—Mordell curves, Ann. Sci. Math. Québec, 281-2:165-178 (2005), 2004  MR 2183104 |  Zbl 1102.11030
[5] Katsuya Miyake, Two expositions on arithmetic of cubics, Number theory, Ser. Number Theory Appl. 2, World Sci. Publ., Hackensack, NJ, 2007  MR 2364840 |  Zbl pre05214171
[6] L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, 1969  MR 249355 |  Zbl 0188.34503
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