Explicit two-cover descent for genus 2 curves

TL;DR

This paper develops explicit methods for two-cover descent on genus 2 curves, enabling the computation of rational points on certain curves by constructing related genus 5 curves and applying elliptic Chabauty.

Contribution

It introduces a new explicit construction of genus 5 curves for two-cover descent on genus 2 curves and implements algorithms in Magma and SageMath for practical computation.

Findings

Successfully computed rational points on 7692 genus 2 curves.

Analyzed success rate and obstacles of the method.

Provided explicit formulas and implementations for descent algorithms.

Abstract

Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases.