X±1(N)X_{\pm 1}(N) (awaiting review)

$X_{\pm 1}(N)$ is the modular curve $X_H$ for $H\le GL_2(\widehat\Z)$ the inverse image of $\begin{pmatrix} \pm 1 & * \\ 0 & * \end{pmatrix} \subset \GL_2(\Z/N\Z)$. As a moduli space it parameterizes pairs $(E,\pm P)$, where:

• $E$ is an elliptic curve over $k$, and
• $P \in E[N]$ is a point of order $N$ with $\pm P$ defined over $k$ (this condition translates to the $x$-coordinate lying in $k$ when $E$ is in short Weierstrass form).

The modular curve $X_1(N)$ is a quadratic refinement of $X_{\pm1}(N)$.