Twists of Hessian Elliptic Curves and Cubic Fields

Miyake, Katsuya. "Twists of Hessian Elliptic Curves and Cubic Fields." Annales mathématiques Blaise Pascal 16.1 (2009): 27-45. <http://eudml.org/doc/10570>.

@article{Miyake2009,
abstract = {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 $\mathbf\{Q\}$.},
affiliation = {Department of Mathematics School of Fundamental Science and Engineering Waseda University 3–4–1 Ohkubo Shinjuku-ku Tokyo, 169-8555 Japan},
author = {Miyake, Katsuya},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Hessian elliptic curves; twists of elliptic curves; cubic fields},
language = {eng},
month = {1},
number = {1},
pages = {27-45},
publisher = {Annales mathématiques Blaise Pascal},
title = {Twists of Hessian Elliptic Curves and Cubic Fields},
url = {http://eudml.org/doc/10570},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Miyake, Katsuya
TI - Twists of Hessian Elliptic Curves and Cubic Fields
JO - Annales mathématiques Blaise Pascal
DA - 2009/1//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 1
SP - 27
EP - 45
AB - 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 $\mathbf{Q}$.
LA - eng
KW - Hessian elliptic curves; twists of elliptic curves; cubic fields
UR - http://eudml.org/doc/10570
ER -