Quadratic Chabauty and rational points I: p-adic heights

TL;DR

This paper advances the explicit determination of rational points on curves by integrating p-adic heights into Kim's nonabelian Chabauty method, surpassing previous approaches like Chabauty-Coleman.

Contribution

It provides the first explicit examples where nonabelian Chabauty, enhanced with p-adic heights, determines rational points beyond classical methods.

Findings

Explicit examples of rational points over Q and quadratic fields.

Demonstration of p-adic heights' role in nonabelian Chabauty.

Extension of Chabauty methods beyond classical limits.

Abstract

We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit nonabelian Chabauty.