Rational points on x3 + x2y2 + y3 = k

Abstract

We study the problem of determining, given an integer k, the rational solutions to [Formula: see text]. For [Formula: see text], the curve [Formula: see text] has genus 3 and its Jacobian is isogenous to the product of three elliptic curves [Formula: see text], [Formula: see text], [Formula: see text]. We explicitly determine the rational points on [Formula: see text] under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all [Formula: see text].