Rational Points of some genus $3$ curves from the rank $0$ quotient strategy

TL;DR

This paper introduces an algorithm to determine all rational points on certain genus 3 curves, specifically those that are degree-2 covers of genus 1 curves with rank 0 Jacobians, extending previous methods to a broader class.

Contribution

It develops and implements a new algorithm for computing rational points on all genus 3 curves that are degree-2 covers of genus 1 curves with rank 0 Jacobians, filling a gap in existing methods.

Findings

Successfully computed rational points for ~40,000 curves

Identified curves meeting Stoll's bound for rational points

Extended Chabauty-Coleman methods to new class of genus 3 curves

Abstract

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of $#C(\mathbb{Q})$ by following Chabauty's approach which considers the special case when the Jacobian variety of $C$ has Mordell-Weil rank $< g$. In 2006, Stoll improved the Coleman's bound. Balakrishnan with her co-authors in [1] implemented the Chabauty-Coleman method to compute the rational points of genus $3$ hyperelliptic curves. Then, Hashimoto and Morrison [8] did the same work for Picard curves. But it happens that this work has not yet been done for all genus 3 curves. In this paper, we describe an algorithm to compute the complete set of rational points $C(\mathbb{Q})$ for any genus $3$ curve $C/\mathbb{Q}$ that is a degree-$2$ cover of a genus $1$ curve whose Jacobian has rank $0$. We implemented this algorithm in Magma, and we ran it on approximately $40, 000$ curves selected from databases of plane quartics and genus $3$ hyperellitic curves. We discuss some interesting examples, and we exhibit curves for which the number of rational points meets the Stoll's bound