The rational points on $X_0\left(N\right)/W_N$ in the case where $N$ is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases $N=91$ and $N=125$ and the $j$-invariants of the corresponding quadratic $\mathbb{Q}$-curves are exhibited.

Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_0^{+}\left(p^r\right)\left(\mathbb{Q}\right)$, for $r\>1$ and $p$ a prime number exceeding $2·{10}^{11}$. This includes the case of the curves $X_{\mathrm{split}}\left(p\right)$. We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11\le p\le {10}^{14}$, $p\ne 13$. The combination of those results completes the qualitative study of rational points on $X_0^{+}\left(p^r\right)$ undertook in our previous work, with the only exception of $p^r={13}^2$.

We derive a relation between induced representations on the group ${GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ which implies a relation between the jacobians of certain modular curves of level $p^2$. The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of ${GL}_2(\mathbb{Z}/p^2\mathbb{Z})$.