Modular curve (reviewed)

For each open subgroup $H \le \GL_2(\widehat\Z)$, there is a modular curve $X_H$, defined as a quotient of the full modular curve $X_{\text{full}}(N)$, where $N$ is the level of $H$. More precisely, $H$ is the inverse image of a subgroup $H_N \le \GL_2(\Z/N\Z)$, which acts on $X_{\text{full}}(N)$ over $\Q$, and $X_H$ is the quotient curve $H_N \backslash X_{\text{full}}(N)$, also defined over $\Q$.

Like $X_{\text{full}}(N)$, the curve $X_H$ is smooth, projective, and integral, and when $\det(H)=\widehat{\Z}$ it is also geometrically integral, but in general it may have several geometric components, as is the case for $X_{\text{full}}(N)$ when $N>2$.

Rational points: When $-1\in H$ the rational points of $X_H$ consist of cusps and $\Gal_\Q$-stable isomorphism classes of pairs $(E,[\iota]_H)$, where $E$ is an elliptic curve over $\Q$, and $[\iota]_H$ is an $H$-level structure on $E$. Such points exist precisely when the image of the adelic Galois representation $\rho_E\colon \Gal_\Q\to \GL_2(\widehat\Z)$ is conjugate to a subgroup of $H$.

Complex points: The congruence subgroup $\Gamma_H:= H\cap \SL_2(\Z)$ acts on the completed upper half-plane $\overline{\mathfrak{h}}$; one connected component of $X_H(\C)$ is biholomorphic to the quotient $\Gamma_H \backslash \overline{\mathfrak{h}}$.

The curve $X_H$ can alternatively be constructed as the coarse moduli space of the stack $\mathcal X_H$ over $\Q$ defined in Deligne-Rapoport [MR:0337993, 10.1007/978-3-540-37855-6_4]. Both constructions of $X_H$ can be carried out over any field of characteristic not dividing $N$, or even over $\Z[1/N]$.