We use the following notation to denote some standard modular curves that appear in the literature:

- $X(N)$ denotes the modular curve that parametrizes elliptic curves with full rational $N$-torsion.
- $X_0(N)$ denotes the modular curve that parametrizes elliptic curves with a rational cyclic isogeny of degree $N$.
- $X_1(N)$ denotes the modular curves that parametrizes elliptic curves with a rational point of order $N$.
- $X_{S_4}(\ell)$ parametrizes elliptic curves whose mod-$\ell$ Galois images is projectively isomorpic to $S_4$, respectively.
- For primes $\ell\equiv 1\bmod 4$ the modular curve $X_{A_4}(\ell)$ parametrizes elliptic curves whose mod-$\ell$ Galois images is projectively isomorpic to $A_4$ and $S_4$, respectively.
- For primes $\ell\equiv\pm 1\bmod 5$ the modular curve $X_{A_5}(\ell)$ parametrizes elliptic curves mod-$\ell$ Galois images is projectively isomorpic to $A_5$.
- $X_{\mathrm{sp}}(N)$ denotes the
**split Cartan**modular curve $X_H$ for which $H\leq \GL_2(\widehat \Z)$ is the inverse image of the reduction to $\GL_2(\Z/N\Z)$ of the Cartan subgroup associated to an imaginary quadratic order $\mathcal O$ in which every prime divisor of $N$ splits (this definition does not depend on $\mathcal O$, up to conjugacy $H$ is the inverse image of the diagonal subgroup of $\GL_2(\Z/N\Z)$). - $X_{\mathrm{sp}}^+(N)$ denotes the modular curve $X_H$ for which $H\leq \GL_2(\widehat\Z)$ is the reduction to $\GL_2(\Z/N\Z)$ of the normalizer of the split Cartan subgroup of $\GL_2(\widehat\Z)$ (containing the split Cartan with index 2).
- $X_{\mathrm{ns}}(N)$ denotes the
**nonsplit Cartan**modular curve $X_H$ for which $H\leq \GL_2(\widehat \Z)$ is the inverse image of the reduction to $\GL_2(\Z/N\Z)$ of the Cartan subgroup associated to an imaginary quadratic order $\mathcal O$ in which every prime divisor of $N$ is inert (this definition does not depend on $\mathcal O$). - $X_{\mathrm{ns}}^+(N)$ denotes the modular curve $X_H$ for which $H\leq \GL_2(\widehat\Z)$ is the reduction to $\GL_2(\Z/N\Z)$ of the normalizer of the nonsplit Cartan subgroup of $\GL_2(\widehat\Z)$ (containing the nonsplit Cartan with index 2).

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2022-03-25 13:20:00

**Referred to by:**

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

- 2022-03-25 13:20:00 by Andrew Sutherland
- 2022-03-24 18:40:04 by Bjorn Poonen
- 2022-03-20 19:57:45 by Andrew Sutherland

**Differences**(show/hide)