## 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 $$t$$.

### MSC:

 11G05 Elliptic curves over global fields 12F05 Algebraic field extensions 14G25 Global ground fields in algebraic geometry
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### References:

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