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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

$\mathbb{Q}(\zeta_N)$.

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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?

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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.

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.

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