Rational points on $X^+_0(125)$

TL;DR

This paper determines all rational points on the curve $X^+_0(125)$ using the quadratic Chabauty method, completing the classification of exceptional rational points on certain modular curves and confirming a conjecture of Galbraith.

Contribution

It applies the quadratic Chabauty method to a specific modular curve, advancing the understanding of rational points on $X^+_0(N)$ for genus 2 to 6.

Findings

All rational points on $X^+_0(125)$ are explicitly determined.

The work completes the classification of rational points on $X^+_0(N)$ for relevant genus ranges.

Confirms Galbraith's conjecture regarding rational points on these curves.

Abstract

We compute the rational points on the Atkin-Lehner quotient $X^+_0(125)$ using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves $X^+_0(N)$ of genus between 2 and 6. Together with the work of several authors, this completes the proof of a conjecture of Galbraith.