Rational points on $X_0(N{)}^{\ast }$ when $N$ is non-squarefree

Let $N$ be a non-squarefree integer such that the quotient $X_0(N{)}^{\ast }$ 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 $X_0(N{)}^{\ast }$ 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 $X_0(N{)}^{\ast }$, 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 $X_0(N{)}^{\ast }$ of genus $1\le g\le 5$, when they are finite. In the process we identify exceptional rational points on $X_0(147{)}^{\ast }$ and $X_0(75{)}^{\ast }$ which were not known before.