Rational points on $X_0(N)^*$ when $N$ is non-squarefree

TL;DR

This paper advances understanding of rational points on certain modular curves by proving integrality of $j$-invariants, classifying rational points for genus 1 to 5, and identifying previously unknown exceptional points, contributing to Elkies' conjecture.

Contribution

It establishes integrality results for rational points on $X_0(N)^*$ and classifies rational points for genus 1 to 5, addressing key cases of Elkies' conjecture.

Findings

Proved integrality of $j$-invariants for non-cuspidal rational points.

Classified rational points on $X_0(N)^*$ for genus 1 to 5.

Identified new exceptional rational points on $X_0(147)^*$ and $X_0(75)^*.

Abstract

Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before.