Twists of Hessian Elliptic Curves and Cubic Fields
Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 27-45.

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

DOI : https://doi.org/10.5802/ambp.251
Classification : 11G05,  12F05
Mots clés : Hessian elliptic curves, twists of elliptic curves, cubic fields
@article{AMBP_2009__16_1_27_0,
author = {Miyake, Katsuya},
title = {Twists of Hessian Elliptic Curves and Cubic Fields},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {27--45},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {16},
number = {1},
year = {2009},
doi = {10.5802/ambp.251},
mrnumber = {2514525},
zbl = {1182.11026},
language = {en},
url = {www.numdam.org/item/AMBP_2009__16_1_27_0/}
}
Miyake, Katsuya. Twists of Hessian Elliptic Curves and Cubic Fields. Annales Mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 27-45. doi : 10.5802/ambp.251. http://www.numdam.org/item/AMBP_2009__16_1_27_0/

[1] Hoshi, Akinari; Miyake, Katsuya Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation, Proc. Japan Acad. Ser. A Math. Sci., Volume 83 (2007) no. 3, pp. 21-26 | Article | MR 2317305 | Zbl 1126.14018

[2] Husemoller, Dale Elliptic curves, Graduate Texts in Mathematics, Volume 111, Springer-Verlag, New York, 1987 (With an appendix by Ruth Lawrence) | MR 868861 | Zbl 0605.14032

[3] Miyake, Katsuya Some families of Mordell curves associated to cubic fields, Proceedings of the International Conference on Special Functions and their Applications (Chennai, 2002), Volume 160 (2003) no. 1-2, pp. 217-231 | MR 2022613 | Zbl 1080.14520

[4] Miyake, Katsuya An introduction to elliptic curves and their Diophantine geometry—Mordell curves, Ann. Sci. Math. Québec, Volume 28 (2004) no. 1-2, p. 165-178 (2005) | MR 2183104 | Zbl 1102.11030

[5] Miyake, Katsuya Two expositions on arithmetic of cubics, Number theory (Ser. Number Theory Appl.) Volume 2, World Sci. Publ., Hackensack, NJ, 2007, pp. 136-154 | MR 2364840 | Zbl pre05214171

[6] Mordell, L. J. Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London, 1969 | MR 249355 | Zbl 0188.34503