$X_0(N)$ 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 $N$.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2023-07-09 08:55:30

**Referred to by:**

**History:**(expand/hide all)

- 2023-07-09 08:55:30 by Andrew Sutherland
- 2022-11-06 17:06:39 by Ciaran Schembri
- 2022-11-06 16:32:01 by Ciaran Schembri
- 2022-11-06 15:30:26 by Ciaran Schembri
- 2022-11-06 15:30:16 by Ciaran Schembri

**Differences**(show/hide)