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
@article{AMBP_2009__16_1_27_0,
     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},
     volume = {16},
     number = {1},
     year = {2009},
     doi = {10.5802/ambp.251},
     mrnumber = {2514525},
     zbl = {1182.11026},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/}
}
TY  - JOUR
AU  - Katsuya Miyake
TI  - Twists of Hessian Elliptic Curves and Cubic Fields
JO  - Annales mathématiques Blaise Pascal
PY  - 2009
DA  - 2009///
SP  - 27
EP  - 45
VL  - 16
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/
UR  - https://www.ams.org/mathscinet-getitem?mr=2514525
UR  - https://zbmath.org/?q=an%3A1182.11026
UR  - https://doi.org/10.5802/ambp.251
DO  - 10.5802/ambp.251
LA  - en
ID  - AMBP_2009__16_1_27_0
ER  - 
%0 Journal Article
%A Katsuya Miyake
%T Twists of Hessian Elliptic Curves and Cubic Fields
%J Annales mathématiques Blaise Pascal
%D 2009
%P 27-45
%V 16
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.251
%R 10.5802/ambp.251
%G en
%F AMBP_2009__16_1_27_0
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/

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

[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

Cited by Sources: