Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

TL;DR

This paper completes the classification of all rational points on certain hyperelliptic Atkin-Lehner quotients of modular curves using advanced Chabauty methods and their coverings.

Contribution

It provides a comprehensive computation of rational points on 64 hyperelliptic Atkin-Lehner quotients and classifies all rational points for square-free levels, including cusps, CM, and exceptional points.

Findings

All rational points on the 64 hyperelliptic quotients are determined.

Classification of rational points for square-free levels into cusps, CM points, and exceptions.

Methodology combining multiple Chabauty techniques and Mordell-Weil sieve.

Abstract

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings.