# Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties

## TL;DR

This paper advances the nonabelian Chabauty method by introducing generalized height functions on Selmer varieties, enabling finiteness proofs of rational points on curves with high Mordell-Weil rank.

## Contribution

It develops a new framework of generalized height functions on Selmer varieties and demonstrates their computation via iterated integrals, leading to explicit nonabelian Chabauty results.

## Key findings

- Proved finiteness of Chabauty--Kim sets in new cases
- Developed a method to compute generalized heights using iterated integrals
- First explicit nonabelian Chabauty result for a curve with Mordell-Weil rank exceeding its genus

## Abstract

We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the first explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell-Weil rank larger than its genus.