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

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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  • $\begingroup$ Is it unfortunate because of the language or because of the authors? :) Better that D-R is going to be hard to do in general, unless you make explicit what is wrong about it! $\endgroup$ – Mariano Suárez-Álvarez Apr 7 '11 at 13:27
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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.

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  • $\begingroup$ Wow, thanks a lot. I will definitely need some time to understand all these concepts, but your notes look helpful! $\endgroup$ – MichalisN Apr 7 '11 at 16:58
  • $\begingroup$ It seems to me that the full modular curve does have an action of the group scheme of "determinant one" automorphisms of $Z/N \times \mu_N$. This seems like a reasonable replacement for $SL_2(Z/N)$ to me. (PS Great answer!) $\endgroup$ – David E Speyer Jul 11 '15 at 16:20
  • $\begingroup$ @David: Yes, I think you can view it as having a nonconstant etale group scheme of automorphisms if you like. $\endgroup$ – Pete L. Clark Jul 11 '15 at 18:21
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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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  • $\begingroup$ Hey matt, sorry for the multiple comment, just in case you missed the first one; I would be really thankful if you extended your answer, or supplied a reference :) $\endgroup$ – MichalisN Apr 8 '11 at 20:56

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