TITLE: The modular curve X(N)
QUESTION [15 upvotes]: I have a question about the modular curve X(N), which classifies elliptic curves with full level N structure. (A level N structure of an elliptic curve E is an isomorphism
from $Z/NZ \times Z/NZ$ to the group of N-torsion points on E).
Some notation: $\Gamma(N)$ is the subgroup of SL$_2(\mathbb{Z})$, which contains all the matrices congruent to the identity matrix modulo N. $\mathbb{H}$ is the upper halfplane.
$\Gamma(N)\backslash\mathbb{H}$ is a Riemannsurface classifying elliptic curves with level N structure and the additional condition that the two base points we choose map to a certain N-th root of unity under the Weil pairing. The problem is that this curve is only defined over
$\mathbb{Q}(\zeta_N)$.
Apparently if we leave out the condition with the Weil pairing, we get a curve X(N) defined over $\mathbb{Q}$ which has $\phi(N)$ geometric components isomorphic to $\Gamma(N)\backslash\mathbb{H}$.
Is there a good way for constructing the curve X(N) from $\Gamma(N)\backslash\mathbb{H}$? Unfortunately the author refers to a french paper by Deligne-Rapoport.(I don't speak French)
Do you know any better references for this?

REPLY [4 votes]: One standard way to describe the disconnected version of $X(N)$ is as follows:
it is the quotient 
$$SL_2(\mathbb Z)\backslash \bigl(\mathcal H \times GL_2(\mathbb Z/N\mathbb Z)\bigr),$$
where $SL_2(\mathbb Z)$ acts on $\mathcal H$ as usual, and on $GL_2(\mathbb Z/N
\mathbb Z)$ via left multiplication of matrices.