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 big modular curve $X(N)^{\text{big}}$, 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 $X(N)^{\text{big}}$ over $\Q$, and $X_H$ is the quotient curve $H_N \backslash X(N)^{\text{big}}$, also defined over $\Q$.

Like $X(N)^{\text{big}}$, the curve $X_H$ is smooth, projective, and integral, but it may have several geometric components and hence not be geometrically integral.

Rational points: If $E$ is an elliptic curve over $\Q$, then each group isomorphism $\beta \colon E[N] \stackrel{\sim}\to (\Z/N\Z)^2$ such that the image of the induced composition $\Gal_\Q \to \Aut E[N] \stackrel{\sim}\to \Aut (\Z/N\Z)^2 = \GL_2(\Z/N\Z)$ is contained in $H_N$ gives a rational point of $X_H$, depending only on the orbit $H_N \beta$. (Given $E$, if we, less canonically, choose an arbitrary basis of $E[N]$ to define $\rho \colon \Gal_\Q \to \GL_2(\Z/N\Z)$, then $\beta$ exists if and only if the image of $\rho$ is contained in a conjugate of $H_N$.) If $-1 \notin H_N$, then conversely, each rational point on $X_H$, excluding the finite set of cusps and possibly finitely many points with $j=0$ or $j=1728$, arises from a unique pair $(E,H_N g)$ up to isomorphism. For fields $k \supset \Q$, the $k$-rational points of $X_H$ can be understood similarly.

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 $\mathcal M_H$ over $\Q$ defined in Deligne-Rapoport [MR:0337993, 10.1007/978-3-540-37855-6_4]. Either construction of $X_H$ can be carried out over any field of characteristic not dividing $N$, or even over $\Z[1/N]$.