$X_0(N)$ (awaiting review)

$X_0(N)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of $\begin{pmatrix} \ast & \ast \\ 0 & \ast \end{pmatrix} \subset \GL_2(\Z/N\Z)$. As a moduli space it parameterizes pairs $(E,C)$, where:

• $E$ is an elliptic curve over $k$, and
• $C$ is a $\Gal_k$-stable cyclic subgroup of $E[N](\overline{k})$ of order $N$ that is the kernel of a rational isogeny $E\to E'$ of degree $N$.