Rational points on X₀(N)^∗ when N is non-squarefree

authored by

Sachi Hashimoto, Timo Keller, Samuel Le Fourn

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. 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 conjecturally complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Preprint

Publication date

01.05.2025

Publication status

E-pub ahead of print

Electronic version(s)

https://doi.org/10.48550/arXiv.2505.00680 (Access: Open)