Rational points on x³ + x² y² + y³ = k

2022/05/26 by Lang, Xiaoan, Rouse, Jeremy
#14H45 #FOS: Mathematics #Number Theory (math.NT) #Primary 11G05 #Secondary 11G30

paper · doi:10.48550/arxiv.2205.13442

Abstract

We study the problem of determining, given an integer k, the rational solutions to C_k : x³z + x² y² + y³z = kz⁴. For k ≠ 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 ∈ ℚ.

Related

• Rational Points on a Family of Genus 3 Hyperelliptic Curves
• Elliptic Curves x³ + y³ = k of High Rank
• Rational Points of some genus 3 curves from the rank 0 quotient strategy
• Rational points in arithmetic progression on y²=xⁿ+k
• Rational points on certain families of symmetric equations