## 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

### MSC:

 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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### References:

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