Twists of Hessian elliptic curves and cubic fields. (English) Zbl 1182.11026

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: \(H_{\mu,t} \to H_\mu\) given by \[ [u:v:w] \mapsto [u+\sqrt{d_t}\,v:u-\sqrt{d_t}\,v:w]. \] Hence, \(\tilde H(\mu,t)\) is isomorphic to \(H_{\mu,t}\) over the splitting field of \(f_t(X)\). In Theorem 5.1, the author uses an explicit map: \(\tilde H(\mu,t) \to H_{\mu,t}\) to prove that the isomorphism is in fact over \(K_t\); if \(f_t\) is reducible over \(\mathbb Q\), then \(K_t\) is defined to be \(\mathbb Q\).
For the case of \(\mu=0\), the author provides the following necessary and sufficient condition for \(\tilde H(\mu,t)\) to have a rational point: \(t=-r^3/(r+1)\) with \(r\in\mathbb Q\) not equal to \(0\) or \(-1\), or \(\tilde H(0,t) \cong \tilde H(0,t')\) where \(t'=h^3/(2h-1)^2\) for some \(h\in\mathbb Q\) not equal to \(0\) or \(1/2\).
In fact, to prove the converse part, the author shows that if \(t'=h^3/(2h-1)^2\) for some \(h\in\mathbb Q\) not equal to \(0\) or \(1/2\), then the curve \(\tilde H(0,t')\) has a rational point, and it is not discussed in the paper whether this can be used to determine whether \(\tilde H(0,t)\) has a rational point for some examples of \(t\).


11G05 Elliptic curves over global fields
12F05 Algebraic field extensions
14G25 Global ground fields in algebraic geometry
Full Text: DOI Numdam EuDML


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