Linear and Quadratic Chabauty for Affine Hyperbolic Curves

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.