---
tag: 55XCZDS142
kind: definition
name: Modular curve
msc: [11G18, 14G35, 11F06]
defines: T5W0CMS0DM
uses: [X6XGGH5FPW, VEKV1EM77B, 1H6R856P3C, 5K0N8QMV6Q]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_curve
---

## Definition

Let $\mathbb{H} = \set{\tau \in \C : \operatorname{Im} \tau > 0}$ be the [[upper half-plane|X6XGGH5FPW]], on which $\mathrm{SL}_2(\Z)$ acts by $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \tau = \frac{a\tau + b}{c\tau + d}$.
Let $\Gamma \subseteq \mathrm{SL}_2(\Z)$ be a [[congruence subgroup|1H6R856P3C]], that is, a subgroup that contains $\Gamma(N) = \set{\gamma \in \mathrm{SL}_2(\Z) : \gamma \equiv I \pmod N}$ for some $N \ge 1$.
The *open modular curve* of $\Gamma$ is the quotient $Y(\Gamma) = \Gamma \backslash \mathbb{H}$.
The *modular curve* of $\Gamma$ is
$$X(\Gamma) = \Gamma \backslash \big( \mathbb{H} \cup \mathbb{P}^1(\Q) \big),$$
where $\mathbb{P}^1(\Q) = \Q \cup \set{\infty}$ is acted on by the same formula; the finitely many points of $X(\Gamma) \setminus Y(\Gamma)$ are the cusps.

## Remarks

With the standard charts at elliptic points and at cusps, $X(\Gamma)$ is a compact [[Riemann surface|5K0N8QMV6Q]], hence a projective algebraic curve over $\C$.
The standard families are $X(N) = X(\Gamma(N))$, $X_0(N)$ for $\Gamma_0(N) = \set{\gamma : c \equiv 0 \pmod N}$ and $X_1(N)$ for $\Gamma_1(N) = \set{\gamma : c \equiv 0,\ a \equiv d \equiv 1 \pmod N}$.
The curves $X_0(N)$ and $X_1(N)$ have models over $\Q$.
The open curve $Y_0(N)$ parametrizes pairs $(E, C)$ of an elliptic curve over $\C$ and a cyclic subgroup $C$ of order $N$, up to isomorphism.

Example: for $\Gamma = \mathrm{SL}_2(\Z)$, the [[modular group|VEKV1EM77B]] up to sign, the $j$-invariant identifies $Y(1)$ with $\C$ and $X(1)$ with $\mathbb{P}^1(\C)$, of genus $0$.
The curve $X_0(11)$ has genus $1$ and is an elliptic curve over $\Q$.

Non-example: a subgroup of finite index in $\mathrm{SL}_2(\Z)$ that contains no $\Gamma(N)$ is not a congruence subgroup.
Its quotient is still a compact Riemann surface, but it is not a modular curve in the sense above.