Linear and quadratic Chabauty for affine hyperbolic curves

TL;DR

This paper establishes conditions for the finiteness of certain Chabauty-Kim loci on affine hyperbolic curves, constructs fundamental group quotients, and provides explicit bounds on S-integral points.

Contribution

It extends the construction of fundamental group quotients to affine hyperbolic curves and applies weight filtrations for explicit bounds on S-integral points.

Findings

Finiteness conditions for linear and quadratic Chabauty-Kim loci.

Construction of depth ≤ 2 fundamental group quotients.

Explicit bounds on S-integral points under certain conditions.

Abstract

We give sufficient conditions for finiteness of linear and quadratic refined Chabauty-Kim loci of affine hyperbolic curves. We achieve this by constructing depth $\leq 2$ quotients of the fundamental group, following a construction of Balakrishnan-Dogra in the projective case. We also apply Betts' machinery of weight filtrations to give unconditional explicit upper bounds on the number of S-integral points when our conditions are satisfied.