Refined Selmer equations for the thrice-punctured line in depth two

TL;DR

This paper refines Kim's method for computing S-integral points on the thrice-punctured line, enabling explicit calculations for cases where S has size 2, thus advancing the practical application of Kim's approach.

Contribution

The paper introduces a refined computational approach to Kim's method, allowing explicit determination of S-integral points for the thrice-punctured line with S of size 2.

Findings

Explicit examples of S-integral points computed for S of size 2

Demonstration of the refined method's effectiveness in practical calculations

Support for a generalized Kim conjecture through new examples

Abstract

In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of $S$ increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where $S$ has size $2$ which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.