Twists of Hessian Elliptic Curves and Cubic Fields
Annales mathématiques Blaise Pascal, Volume 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: 10.5802/ambp.251
Classification: 11G05, 12F05
Keywords: Hessian elliptic curves, twists of elliptic curves, cubic fields
Katsuya Miyake 1

1 Department of Mathematics School of Fundamental Science and Engineering Waseda University 3–4–1 Ohkubo Shinjuku-ku Tokyo, 169-8555 Japan
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Katsuya Miyake. Twists of Hessian Elliptic Curves and Cubic Fields. Annales mathématiques Blaise Pascal, Volume 16 (2009) no. 1, pp. 27-45. doi : 10.5802/ambp.251. https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/

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[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), Volume 160 (2003) no. 1-2, pp. 217-231 | MR | Zbl

[4] Katsuya Miyake 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 | Zbl

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

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

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