Twists of Hessian Elliptic Curves and Cubic Fields

Miyake, Katsuya

Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 27-45 / Harvested from Numdam

Résumé

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 $\mathbf{Q}$.

Détail

Publié le : 2009-01-01

MR 2514525 | Zbl 1182.11026

DOI : https://doi.org/10.5802/ambp.251
Classification:  11G05,  12F05

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