show · modcurve.x0 all knowls · up · search:

$X_0(N)$ 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 $N$.

Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2023-07-09 08:55:30
Referred to by:
History: (expand/hide all) Differences (show/hide)