Examples of effectivity for integral points on certain curves of genus 2

TL;DR

This paper investigates the effective determination of integral points on certain genus 2 curves with a missing point, using covers and torsion analysis to identify fibers with computable integral points.

Contribution

It introduces a method involving degree-3 étale covers and torsion sections to effectively find integral points on families of genus 2 curves.

Findings

Dense set of fibers with effectively computable integral points

Construction of degree-3 étale covers for genus 2 curves

Analysis of torsion values of sections in doubly elliptic abelian schemes

Abstract

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively determined. Our method is based on the construction of degree-3 \'etale covers of such curves of genus 2 and the study of the torsion values of sections of certain doubly elliptic abelian schemes.