Affine Chabauty II

TL;DR

This paper introduces an algorithm for finding $S$-integral points on affine curves using the Affine Chabauty method, involving explicit logarithmic differentials and a new $p$-adic residue theorem for Coleman integrals.

Contribution

It develops a novel algorithm based on explicit logarithmic differentials and proves a new $p$-adic residue theorem to enhance the Affine Chabauty method.

Findings

Algorithm successfully determines $S$-integral points on affine curves.

Constructs explicit logarithmic differentials with prescribed integral values.

Proves a $p$-adic residue theorem for Coleman integrals of log differentials.

Abstract

We present an algorithm for determining the set of $S$-integral points on an affine curve based on the Affine Chabauty method developed in the first part of this series. We achieve this by constructing explicit logarithmic differentials whose integrals take on prescribed values on $S$-integral points. Along the way, we prove a $p$-adic residue theorem for Coleman integrals of log differentials.