A family of elliptic curves and cyclic cubic field extensions. (English) Zbl 0555.14010

Let K be a field, \(char K\neq 2,3.\) The problem here is to find rational points over K on a family of elliptic curves \(F_{\lambda}/K\), where \(F_{\lambda}\) are defined by \(x^ 3+y^ 3+z^ 3=\lambda xyz,\) \(\lambda \in K,\quad \lambda^ 3\neq 27.\) The principle employed is to relate points on \(F_{\lambda}\) with that on another family of projective elliptic curves \(E_{\lambda}\), where \(E_{\lambda}\) are defined by \(u+v+w=\lambda\), \(uvw=1\) plus the points at infinity \(e_ 1\) (origin), \(e_ 2\), \(e_ 3.\)
A point \(p\in E_{\lambda}\) is called a cyclic cubic point if p is a K- rational point or p(u,v,w) with conjugate u,v,w in a cyclic cubic extension of K. - Let \(E_{\lambda}(C;K)\) be the set of all cyclic cubic points on \(E_{\lambda}\). Then the main theorem asserts that there exists a rational map \(\rho: E_{\lambda}\to F_{\lambda}\) with \(Ker \rho =\{e_ 1,e_ 2,e_ 3\}\) and \(E_{\lambda}(C;K)=\rho^{- 1}F_{\lambda}(K)\) \((F_{\lambda}(K)\) is the set of all K-rational points on \(F_{\lambda})\). The author applies the main theorem to find an infinite sequence \(\{\lambda_ n\}\), \(\lambda_ n\in {\mathbb{Q}}\), such that rank \(F_{\lambda_ n}({\mathbb{Q}})>0.\)
Reviewer: K.Katayama


14G05 Rational points
14H45 Special algebraic curves and curves of low genus
11R16 Cubic and quartic extensions
14H52 Elliptic curves
14H10 Families, moduli of curves (algebraic)
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