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

TL;DR

This paper demonstrates a method to effectively find integral points on certain genus 2 curves using Bilu's criterion, illustrating examples with increasing complexity and field of definition.

Contribution

It applies Bilu's criterion to specific genus 2 curves, providing explicit examples and constructing morphisms with increasing degrees to show effectivity in integral point search.

Findings

Effective method for integral points on genus 2 curves demonstrated

Construction of morphisms with increasing degrees to illustrate complexity

Examples become more interesting with higher degrees and larger fields of definition

Abstract

This short article concerns a method to obtain effectivity for the search of integral points on certain (sets of) curves of genus 2. More precisely, we wish to illustrate just an example of application of a criterion of Bilu, to derive effectivity for integral points on certain families of affine curves of genus 2. Future work, in collaboration with D. Lombardo, will contain details and more general applications. We shall construct morphisms from these curves to $\G_{\rm m}^2$, with image of increasing degrees. We note that as the degree increases, we may say that the examples become `more interesting', since they cannot be derived by substitution from a universal family. As a counterpart, there is the negative feature in that the relevant curves will somewhat have increasing fields of definition.