Rational points on x3 + x2y2 + y3 = k

Abstract

We study the problem of determining, given an integer $k$, the rational solutions to $C_k:x^{3}z+x^2{y}^2+y^{3}z=k{z}^4$. For $k\ne 0$, the curve $C_k$ has genus $3$ and its Jacobian is isogenous to the product of three elliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine the rational points on $C_k$ under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all $k\in \mathbb{Q}$.