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$X_{\pm 1}(N)$ 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 $X_{\pm1}(N)$.

Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2023-07-09 08:57:26
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