$X_1(N)$ (awaiting review)

$X_1(N)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of $\begin{pmatrix} 1 & * \\ 0 & * \end{pmatrix} < \GL_2(\Z/N\Z)$. As a moduli space it parameterizes pairs $(E,P)$, where $E$ is an elliptic curve over $k$, and $P \in E[N](k)$ is a point of exact order $N$.