The modular curve X(N) 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?
 A: 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.
A: The way you construct $X(N)_{/\mathbb{Q}}$ from $\Gamma(N) \backslash \mathcal{H}$ is exactly as you said: $\Gamma(N) \backslash \mathcal{H}$ is by its nature a complex-analytic object, more precisely a Riemann surface which is the complex points of an affine algebraic curve over $\mathbb{C}$.  To get from this to what you call $X(N)$ one comes up with a moduli problem: elliptic curves, plus $\Gamma(N)$-level structure.
Because a certain mod $N$ determinant map fails to be surjective in the case of $\Gamma = \Gamma(N)$ (see page 11 of these notes for more information on this) the canonical model in the sense of Shimura and Deligne is defined over a proper extension of $\mathbb{Q}$, in this case $\mathbb{Q}(\zeta_N)$.  This corresponds to the fact that the moduli problem has a nontrivial discrete invariant: a choice of primitive $N$th root of unity.  So the geometrically disconnected curve $X(N)$ is one way of "pushing the moduli problem down to $\mathbb{Q}$".  I don't see how to describe $X(N)$ in terms of the complex algebraic curve in any more direct way than this.
You should also be aware that $\mathbb{Q}$ is a field of definition of the complex algebraic curve $\Gamma(N) \backslash \mathcal{H}$, i.e., there is an algebraic curve over $\mathbb{Q}$ whose base extension to $\mathbb{C}$ is the given curve.  One way to show this is to use the arithmetic theory of branched coverings.  This perspective is applied to a more general family of curves here.  Or it can be done by twisting the moduli problem: rather than considering elliptic curves with full $N$-torsion rational over the ground field, one can consider elliptic curves $E$ over a field $K$ with $N$-torsion subgroup scheme is Weil-equivariantly isomorphic to $\mathbb{Z}/N /\mathbb{Z} \times \mu_N$.  One thing that one loses here is the automorphisms: the complex analytic curve has $\operatorname{PSL}_2(\mathbb{Z}/N\mathbb{Z})$ acting on it by automorphisms (in fact this is the full automorphism group for all sufficiently large $N$).  But there is no model over $\mathbb{Q}$ for which all these automorphisms
are $\mathbb{Q}$-rationally defined.
By the way, if there is any royal road to these results, I am not aware of it.  If you are serious about understanding this material, you will eventually need to read (at least parts of) Deligne-Rapoport, French or no.  In fact you will probably find that the linguistic issues are the least of your worries: there are other closely related works by Shimura and Katz-Mazur, all in English, but these works all employ certain mathematical dialects (e.g. Weil-style foundations, moduli stacks, fppf topologies) that take time to learn to speak as well.  In my memory at least, Deligne-Rapoport is very clearly written and uses close to the minimum possible amount of technology necessary to cover the topics that appear therein: i.e., a reasonable command of algebraic and arithmetic geometry in the language of schemes (which is sort of Esperanto for mathematicians having some interest in algebraic geometry, these days).  Good luck.
