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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

URL stable: http://www.numdam.org/item?id=AMBP_2009__16_1_27_0

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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 $\mathbf{Q}$.

Bibliographie

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