Twists of Hessian Elliptic Curves and Cubic Fields
Miyake, Katsuya1
1 Department of Mathematics School of Fundamental Science and Engineering Waseda University 3–4–1 Ohkubo Shinjuku-ku Tokyo, 169-8555 Japan
Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 27-45.

In this paper we investigate Hesse’s elliptic curves H μ :U 3 +V 3 +W 3 =3μUVW,μ∈Q-{1}, and construct their twists, H μ,t over quadratic fields, and H ˜(μ,t),μ,t∈Q over the Galois closures of cubic fields. We also show that H μ is a twist of H ˜(μ,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∈Q-{0,-27/4}, to parametrize all of quadratic fields and cubic ones. It should be noted that H ˜(μ,t) is a twist of H μ as algebraic curves because it may not always have any rational points over Q. We also describe the set of Q-rational points of H ˜(μ,t) by a certain subset of the cubic field. In the case of μ=0, we give a criterion for H ˜(0,t) to have a rational point over Q.

DOI : 10.5802/ambp.251
Classification : 11G05, 12F05
Mots-clés : Hessian elliptic curves, twists of elliptic curves, cubic fields
Miyake, Katsuya 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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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/articles/10.5802/ambp.251/

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | Zbl

[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

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

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