Chabauty without the Mordell-Weil group

TL;DR

This paper introduces a new algorithm for finding rational points on algebraic curves using only the p-Selmer group, especially effective for high-genus curves where traditional methods struggle.

Contribution

It presents a novel approach that bypasses the need for the Mordell-Weil group, expanding the applicability of Chabauty's method to more complex curves.

Findings

Successfully applied to generalized Fermat equations

Effective for high-genus curves with limited Mordell-Weil data

Provides a new tool for rational point determination

Abstract

Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve $C$, given only the $p$-Selmer group $S$ of its Jacobian (or some other abelian variety $C$ maps to) and the image of the $p$-Selmer set of $C$ in $S$. The method is more likely to succeed when the genus is large, which is when it is usually rather difficult to obtain generators of a finite-index subgroup of the Mordell-Weil group, which one would need to apply Chabauty's method in the usual way. We give some applications, for example to generalized Fermat equations of the form $x^5 + y^5 = z^p$.