Twists of Hessian elliptic curves and cubic fields
type: Item
instance of: scholarly article
title: Twists of Hessian elliptic curves and cubic fields
zbMATH Open document ID: 1182.11026
author: Katsuya Miyake
published in: Annales Mathématiques Blaise Pascal
publication date: 2009-05-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10570
review text: The paper concerns the family of elliptic curves considered by Hesse in the 1840's: \[ H_\mu : U^3 + V^3 + W^3 = 3\mu U V W \] where \(\mu\neq 1\). The author's main result is the construction of special twists of \(H_\mu\), which are defined as follows: Let \(\Xi\) be the matrix \[ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1\\ -t & -t & 0 \end{pmatrix}, \] and let \(M\) be the matrix \(x I_3 + y\,\Xi + z\,\Xi^2\) where \(x\), \(y\) and \(z\) are real-valued indeterminates. Let \(\tilde H(\mu,t)\) denote the projective curve given by \[ \text{Tr}( M^3 ) = 3\mu\, \det( M ). \] The author proves that \(\tilde H(\mu,t)\) is a twist of \(H_\mu\) as curves over \(\mathbb Q\) if \(\mu\neq 1\) and \(t \neq 0,-27/4\) are rational numbers, and that the two curves become isomorphic over the splitting field of \(f_t(X):=X^3+tX+t\), which is the characteristic polynomial of \(\Xi\). The way the curve \(\tilde H(\mu,t)\) is constructed immediately allows the following description of the set of rational points on \(\tilde H(\mu,t)\) if \(f_t(X)\) is irreducible over \(\mathbb Q\): Let \(f_t\) be irreducible over \(\mathbb Q\), let \(K_t\) be the field \(\mathbb Q[X]/(f_t(X))\), and let \(S\) be the group \[ \{\eta \in K_t^* : \text{Tr}_{K_t/\mathbb Q}( \eta^3 ) = 3\mu\, \text{N}_{K_t/\mathbb Q}(\eta) \} \] on which \(\mathbb Q^*\) naturally acts via multiplication. The author establishes a canonical one-to-one correspondence between the set of orbits in \(S\) under this action and the set of rational points on \(\tilde H(\mu,t)\). Another family \(\{H_{\mu,t}\}\) of twists of \(H_\mu\) is introduced in the paper, which is defined by \[ H_{\mu,t} : 2u^3 + 6d_t uv^2 + w^3 = 3\mu(u^2-d_tv^3) w \] where \(d_t=-(4t+27)\); the field \(\mathbb Q(\sqrt{d_t})\) is an intermediate field of the splitting field of \(f_t(X)\). It is trivial that for \(\mu\neq 1\) and \(t \neq 0,-27/4\), the curve \(H_{\mu,t}\) is an elliptic curve and it is a quadratic twist of \(H_\mu\) as there is a map: \(H_{\m…
reviewed by: Sungkon Chang
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 5554657
zbMATH Keywords: Hessian elliptic curves
zbMATH Keywords: twists of elliptic curves
zbMATH Keywords: cubic fields
MaRDI profile type: Publication
OpenAlex ID: W1980850432
cites work: Tschirnhausen transformation of a cubic generic polynomial and a \(2\)-dimensional involutive Cremona transformation
cites work: Q3743417
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cites work: Q5477595
cites work: TWO EXPOSITIONS ON ARITHMETIC OF CUBICS
cites work: Q5582375
DOI: 10.5802/AMBP.251
description: scientific article; zbMATH DE number 5554657