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 μ :U 3 +V 3 +W 3 =3μUVW,μQ-{1}, and construct their twists, H μ,t over quadratic fields, and H ˜(μ,t),μ,tQ 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,tQ-{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
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
     author = {Katsuya Miyake},
     title = {Twists of {Hessian} {Elliptic} {Curves} and {Cubic} {Fields}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {27--45},
     publisher = {Annales math\'ematiques Blaise Pascal},
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     year = {2009},
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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.

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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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