Mixed Tate motives and the unit equation II

TL;DR

This paper advances Kim's fundamental group approach to effectively compute integral points on hyperbolic curves, specifically focusing on the thrice punctured line, through an algorithm that relates to deep conjectures.

Contribution

It constructs an algorithm for determining integral points on the thrice punctured line, moving towards effective solutions over the rationals and exploring higher number fields.

Findings

Developed an algorithm for integral points on hyperbolic curves.

Proved the algorithm's correctness assuming certain conjectures.

Extended the framework to higher number fields.

Abstract

Over the past fifteen years or so, Minhyong Kim has developed a framework for making effective use of the fundamental group to bound (or even compute) integral points on hyperbolic curves. This is the third installment in a series whose goal is to realize the potential effectivity of Kim's approach in the case of the thrice punctured line. As envisioned in the last installment, we construct an algorithm whose output upon halting is provably the set of integral points, and whose halting would follow from conjectures. Our results go a long way towards achieving our goals over the rationals, while broaching the topic of higher number fields.