Modular curve

Remarks

With the standard charts at elliptic points and at cusps, $X(\Gamma)$ is a compact Riemann surface, 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 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.