Intermediate modular curves with infinitely many quartic points

Abstract

For every group $\{\pm 1\}\subseteq \hat{I}”\subseteq (\mathbb{Z}/N\mathbb{Z}{)}^{\times }$, there exists an intermediate modular curve $X_{\hat{I}}”(N)$. In this paper we determine all curves $X_{\hat{I}}”(N)$ with infinitely many points of degree $4$ over $\mathbb{Q}$. To do that, we developed a method to compute possible degrees of rational morphisms from $X_{\hat{I}}”(N)$ to an elliptic curve.