Rational points on \(x^3+ x^2 y^2+ y^3=k\) (Q6594429)

Let \(k\ne 0,\in\mathbb Z\) and \(C_k\) be the curve defined by \(x^3+x^2y^2+y^3=k\). In this article, the rational points \(C_k(\mathbb Q)\) of \(C_k\) are determined. The Jacobian of \(C_k\) is of genus \(3\) and is isogenous to the product of three elliptic curves \(E_{i,k} (i=1,2,3)\). The authors explicitly give the equation of \(E_{i,k}\) and the morphism \(\phi_i\) of \(C_k\) to \(E_{i,k}\) for each \(i\). Since \(C_k(\mathbb Q)\subset \phi_i^{-1}(E_{i,k}(\mathbb Q))\), assuming that one of \(E_{i,k}(\mathbb Q)\) is of rank \(0\), they show that the only rational points of \(C_k\) are ``obvious'' ones. To determine the torsion groups of \(E_{i,k}(\mathbb Q)\), they use Mazur's result on rational torsion groups of elliptic curves over \(\mathbb Q\) and Kubert's universal elliptic curve with a point of order \(n\), for \(n\le 7\). They find a curve \(C_{i,n}\) that parametrizes \((P,k)\) for which \(P\in C_k(\mathbb Q)\) and \(\phi_i(P)\in E_{i,k}(\mathbb Q)\) has order \(n\) and a curve \(D_{i,n}\) parameterizing values of \(k\) for which \(E_{i,k}(\mathbb Q)\) contains a point of order \(n\). The results follow from determination of the integer values of \(k\) giving a rational point on \(C_{i,k}\). Since there exists a map from \(C_{i,n}\rightarrow D_{i,n}\), to this purpose, it is sufficient to analyze \(D_{i,n}\) if \(D_{in}\) has only finitely many rational points, for example, the genus of \(D_{i,n}\ge 2\). However, if \(C_{i,n}\) and \(D_{i,n}\) have infinitely many rational points, then the authors use Lemma 2.1. to determine \(C_k(\mathbb Q)\).