Elliptic curves related to cyclic cubic extensions. II. (Q2866581)

Let \(F\) be a field and \(K/F\) a cubic Galois extension with basis \(\{1,\alpha_1,\alpha_2\}\) and \(\operatorname{Gal}(K/F)=\langle\sigma\rangle\). Let \(x,y,z,w\) be elements algebraically independent over \(F\) and consider the surface (in \(\mathbb{P}^3\)) \(\mathcal{X}:\;N(x+y\alpha_1+z\alpha_2)=w^3\) (where \(N:K\rightarrow F\) is the norm map). The paper studies the set \(\mathcal{E}(\mathcal{X})\) of isomorphism classes of elliptic curves which arise as \(\mathcal{X}_h:=\mathcal{X}\cap h\) for some plane \(h\) of \(\mathbb{P}^3\). The main results are generalizations and improvements of a previous paper [the author, Int. J. Number Theory 5, No. 4, 591--623 (2009; Zbl 1196.11078)] in which only the number field case was treated. In particular, mainly by means of direct computation and a careful analysis of all possible cases, the author shows that each class \([\mathcal{X}_h]\) has a representative isomorphic to a curve with equation NEWLINE\[NEWLINE y^2z+\operatorname{Tr}(\gamma_h)xyz+N(\gamma_h)yz^2=x^3, NEWLINE\]NEWLINE where \(\operatorname{Tr}:K\rightarrow F\) is the trace map, \(h\) has the form \(a(x-w)+by+cz=0\) and \(\gamma_h:=\det\begin{pmatrix} 1 & \alpha_1 & \alpha_2 \\ 1 & \alpha_1^\sigma & \alpha_2^\sigma \\ a & b & c \end{pmatrix}\), and that the set \(\mathcal{E}(\mathcal{X})\) is equal to the set \(\mathcal{E}(K/F)\) of isomorphism classes of elliptic curves \(E_{/F}\) such that{\parindent=0.7cm \begin{itemize}\item[(i)] there is \(T\in E[3](F)-O\); \item[(ii)] let \(\varphi:E\rightarrow \widehat{E}:=E/\langle T\rangle\), then exists \(P\in \widehat{E}(F)\) such that \(K=F(\varphi^{-1}(P))\). NEWLINENEWLINE\end{itemize}}