An effective Chabauty-Kim theorem

TL;DR

This paper extends the Chabauty--Kim method to higher rank curves, providing a nonabelian generalization of Coleman's effective Chabauty theorem for bounding rational points.

Contribution

It demonstrates how the Chabauty--Kim method can be applied in depth 2 to bound rational points on higher rank curves, generalizing Coleman's theorem.

Findings

Bound the number of rational points on higher rank curves

Extend Chabauty--Kim method to depth 2 cases

Generalize Coleman's effective Chabauty theorem

Abstract

The Chabauty--Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves with Mordell--Weil rank less than their genus. We show how the Chabauty--Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a nonabelian generalisation of Coleman's effective Chabauty theorem.