Rational points on $x^{3} + x^{2} y^{2} + y^{3} = k$

TL;DR

This paper investigates rational solutions to a specific genus 3 curve, utilizing elliptic curve mappings, and explicitly determines solutions for cases where associated elliptic curves have rank zero, highlighting challenges in generalization.

Contribution

It provides explicit rational point determinations on the curve for cases with rank-zero elliptic curves, advancing understanding of rational solutions on genus 3 curves.

Findings

Explicit rational points determined for certain $k$

Method relies on elliptic curve rank conditions

Discussion of extending results to all $k$

Abstract

We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to 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 $k \in \mathbb{Q}$.