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Let $\Gamma\subset {SL}_2(\mathbb Z)$ be a congruence subgroup, and let $X(\Gamma)$ be the associated compact modular curve. The inclusion $\Gamma \subset SL_2(\mathbb Z)$ induces a canonical projection $p:X(\Gamma)\to X(1)$ which is a non-constant holomorphic map of compact Riemann surfaces, where $X(1)=X(SL_2(\mathbb Z))$. The map $p$ ramifies at most over the elliptic points $i=\sqrt{-1}$ and $\mu_3=e^{2\pi i/3}$ of $X(1)$ and over the cusp $\infty$ of $X(1)$. Let $d$ be the degree of $p$ and let $\overline{\mathbb Q}$ be an algebraic closure of $\mathbb Q$.

Assumption $(*)$: There exists a smooth, projective and connected curve $X(\Gamma)_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that its base change to $\mathbb C$ corresponds to $X(\Gamma)$.

  1. If assumption $(*)$ is satisfied, does there exists a non-constant morphism $X(\Gamma)_{\overline{\mathbb Q}}\to \mathbb P^1$ of algebraic curves over $\bar{\mathbb Q}$ of degree $d$, which ramifies at most over $i$, $\mu_3$ and $\infty$ (viewed as points of $\mathbb P^1=X(1)_{\overline{\mathbb Q}}$)?

  2. Is assumption $(*)$ always satisfied?

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Yes and yes. You don't deduce 2 from 1, you prove them both at the same time by proving that if $\Gamma=\Gamma(N)$ is the principal congruence subgroup, then there's a non-compactified moduli space representing elliptic curves over $\overline{\mathbf{Q}}$ with two points of order $N$ generating the $N$-torsion; then you let $Y(\Gamma)$ be one of the components of this moduli space and let $X(\Gamma)$ be the compactification. Both (1) and (2) now follow; to deal with general $\Gamma$ you just quotient out $X(\Gamma(N))$ by the finite group $\Gamma/\Gamma(N)$. Done! –  user30035 Mar 14 '13 at 22:45
meh should say "don't deduce 1 from 2". Roll on the days when I can edit comments. –  user30035 Mar 14 '13 at 22:46
Dear wccanard, thanks for the answer. –  Guillaume Pastorini Mar 15 '13 at 8:50

1 Answer 1

up vote 1 down vote accepted

Let $X(\Gamma)\to \mathbf{P}^1_{\mathbf C}$ be the composition of the natural map $X(\Gamma)\to X(1)$ associated to the inclusion $\Gamma\subset$ SL$_2(\mathbf Z)$, and the $j$-invariant $j:X(1)\to \mathbf{P}^1_{\mathbf C}$. This map is, as you said, ramified over at most three points. Since these three points are algebraic numbers, by a theorem of Grothendieck and Weil, there exists a unique pair $(Y,Y\to \mathbf{P}^1_{\overline{\mathbf Q}})$ with $Y$ a smooth projective connected curve over $\overline{\mathbf Q}$ and $Y\to \mathbf{P}^1_{\overline{\mathbf Q}}$ a finite morphism which gives the natural map $X(\Gamma)\to \mathbf{P}^1_{\mathbf C}$ after base-change. If you just want a model for your $X(\Gamma)$ over $\overline {\mathbf Q}$ as a curve (and not as a cover) you have more choice.

The theorem of Grothendieck and Weil is more general. Let $k\subset K$ be an extension of algebraically closed fields of characteristic zero. Then, for any smooth quasi-projective connected variety $U$ over $k$, the base-change functor is an equivalence of categories from "finite etale covers of $U$" to "finite etale covers of $U_K$". We are applying this theorem to $\mathbf P^1_{k} -\{0,1,\infty\}$ with $k=\overline {\mathbf Q}$ and $K=\mathbf C$.

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Dear Ariyan Javanpeykar. Thank you very much for your excellent answer. Just to make sure that I understand your arguments completely: 1. The non-constant holomorphic map $X(\Gamma)\to X(1)$ of compact Riemann surfaces induces a non-constant morphism $X(\Gamma)\to X(1)$ of algebraic curves over $\mathbb C$, with the same degree and the same ramification points? 2. What is the etale cover $V\to U=\mathbb P^1-\{0,1,\infty\}$ to which you are applaying G-W (is $V=(X(\Gamma)-\{points \ above \ 0,1,\infty\})$ and then you restrict the natural map)? –  Guillaume Pastorini Mar 15 '13 at 9:28
1. Yes. 2. V is the inverse image of U as you already suspected. Let me know if anything else seems obscure. –  Ari Mar 16 '13 at 20:40
This answer is certainly correct, but wccanard makes an important point: the viewpoint of the original question may be "mistaken" (depending on the motivation). The relationship between certain maps among algebro-geometric moduli spaces and maps between their "uniformized analytifications" rests on arguments that have a lot to do with GAGA and moduli functors (and nothing to do with Grothendieck-Weil). For many applications in number theory (work of Mazur, Gross-Zagier, etc.), one needs this alternative viewpoint in order to work with points over specific number fields, etc. –  user30379 Mar 17 '13 at 5:48
@pranavk You are absolutely right. The point wccanard makes is certainly important. The above point of view doesn't take any moduli interpretation into account which Guillaume probably wants. I recommend Guillaume to keep my answer in mind (also when dealing with certain higher-dimensional Shimura varieties), but to always try and use the moduli-interpretation as much as possible as wccanard does. It's more natural when working with modular curves. –  Ari Mar 17 '13 at 15:03

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