Two recent p-adic approaches towards the (effective) Mordell conjecture

TL;DR

This paper introduces two recent p-adic methods, including the non-abelian Chabauty approach, for effectively proving the Mordell conjecture and determining rational points on certain curves.

Contribution

It provides an accessible overview of Lawrence--Venkatesh and Kim's approaches, highlighting the effective application of the non-abelian Chabauty method with new examples.

Findings

Effective determination of rational points on modular curves

Illustration of non-abelian Chabauty method in practice

Three new examples of curves with explicitly known rational points

Abstract

We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.