Geometric quadratic Chabauty

TL;DR

This paper refines the quadratic Chabauty method for finding rational points on curves of genus at least 2, making it more geometric and algebraic, and potentially more accessible and applicable.

Contribution

It reformulates the quadratic Chabauty method using basic algebraic geometry, simplifying its conceptual framework and broadening its potential applications.

Findings

Reformulation of quadratic Chabauty in algebraic geometric terms

Application to specific curves with rank equal to genus

Enhanced understanding of the geometric structure of the method

Abstract

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only `simple algebraic geometry' (line bundles over the jacobian and models over the integers).