Affine Chabauty I

TL;DR

This paper develops an affine Chabauty method to prove finiteness and explicitly bound the number of $S$-integral points on certain affine curves, extending classical techniques with new arithmetic intersection bounds.

Contribution

It introduces an affine Chabauty approach, embedding curves into generalized Jacobians and bounding $S$-integral points using arithmetic intersection theory, with foundations for a computational method.

Findings

Proves finiteness of $S$-integral points under a rank-genus inequality.

Provides explicit upper bounds for the number of $S$-integral points.

Establishes a framework for computational determination of $S$-integral points.

Abstract

We prove finiteness and give an explicit upper bound on the number of $S$-integral points on affine curves satisfying a certain rank-genus inequality. We achieve this by developing an analogue of the Chabauty method, embedding the curve into its generalised Jacobian and bounding the Abel-Jacobi image of the $S$-integral points using arithmetic intersection theory. Our results also provide the foundations for a computational method to determine the set of $S$-integral points on affine curves which will be presented in a follow-up article.