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Consider the familiar Riemann surface

$$ Y_1(N) = \Gamma_1(N) \backslash \mathcal{H} $$

where $\mathcal{H}$ is the upper half-plane and $\Gamma_1(N)$ is the subgroup of matrices in $SL_2(\mathbb{Z})$ which are congruent to $\begin{pmatrix} 1 & * \\\ 0 & 1 \end{pmatrix}$ modulo $N$.

It's a standard theorem that $Y_1(N)$ has a canonical model as an algebraic curve over $\mathbb{Q}$, and this model is a moduli space for pairs $(E, P)$ where $E$ is an elliptic curve and $P$ is an point of order $N$, with the map from $\Gamma_1(N) \backslash \mathcal{H}$ given by sending $\tau$ to $(\mathbb{C} / (\mathbb{Z} + \mathbb{Z} \tau), 1/N)$,

I used to believe that the function field of this canonical $\mathbb{Q}$-model was exactly the meromorphic $\Gamma_1(N)$-invariant functions (with sufficiently slow growth at the cusps) whose $q$-expansions at $\infty$ have coefficients in $\mathbb{Q}$. But some stuff I've just read on Siegel units convinces me that this can't be true.

  • Can one characterize the rational functions on the canonical $\mathbb{Q}$-model in terms of $q$-expansions?
  • Does the field of modular functions with rational $q$-expansions also give a model of $Y_1(N)$ over $\mathbb{Q}$? If so, does it have any natural interpretation as a moduli space?
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In the model you describe, the cusp $\infty$ of $X_1(N)$ is not defined over ${\bf Q}$ (but the cusp $0$ is). A way to see this is that the marked elliptic curve $({\bf C}/({\bf Z}+\tau{\bf Z}),1/N)$ is isomorphic to the marked Tate curve $E_q=({\bf C}^\times/q^{\bf Z},e^{2\pi i/N})$ with $q=e^{2\pi i\tau}$. When you let $\tau \to \infty$, you get $q \to 0$ so that $E_q \to ({\bf G}_m,e^{2i\pi/N})$, which is not defined over ${\bf Q}$. This fact is explained in Diamond-Im, Modular forms and modular curves, see 9.3.5 and 9.3.6.

There is an alternative model $Y_\mu(N)$ classifying elliptic curves $E$ together with a closed immersion $\mu_N \hookrightarrow E$ (see loc. cit. 8.2.2). In this model the cusp $\infty$ is defined over ${\bf Q}$, so it gives an affirmative answer to your second question.

You can switch from one model to another with the Atkin-Lehner involution $W_N$, which becomes an isomorphism defined over ${\bf Q}$ — it is only defined over ${\bf Q}(\mu_N)$ when considered as an involution of either $X_1(N)$ or $X_{\mu}(N)$. But I don't see a nice way to characterize those functions which are rational for the canonical model in terms of the $q$-expansion at $\infty$.

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    $\begingroup$ I think your $Y_\mu(N)$ is a model for $Y_0(N)$. For one thing, $Y_1(N)$ parametrizes elliptic curves with a point of exact order N, not with a cyclic subgroup of order $N$ as $\mu_N$ is. The two moduli problems for $Z/N$ embeddings and $\mu_N$ embeddings are the $\mathcal{A}_N$ and $\mathcal{B}_N$ moduli problems in Deligne-Rapoport's section V.1. $\endgroup$
    – stankewicz
    Commented Nov 16, 2012 at 21:45
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    $\begingroup$ @stankewicz : I don't think the $Y_\mu(N)$ described here is a model for $Y_0(N)$. I really consider the closed immersion as part of the data, not only its image in $E$, like the $\mathcal{A}_N$ and $\mathcal{B}_N$ moduli problems in Deligne-Rapoport. Two closed immersions $i_1 : \mu_N \to E_1$ and $i_2 : \mu_N \to E_2$ are equivalent when there is an isomorphism $psi : E_1 \xrightarrow{\cong} E_2$ such that $i_2 = \psi \circ i_1$. Any $\alpha \in ({\bf Z}/N{\bf Z})^\times$ defines an isomorphism $[\alpha] : \mu_N \to \mu_N$, but in general $(E,i) \not\cong (E,i \circ [\alpha]$. $\endgroup$
    – François Brunault
    Commented Nov 17, 2012 at 9:10
  • $\begingroup$ If I've understood Deligne--Rapoport correctly, if $N$ is invertible on the base the moduli problems $\mathcal{A}_N$ and $\mathcal{B}_N$ are identical, aren't they? When $N$ is invertible, $\mu_N$ is etale-locally isomorphic to $\mathbb{Z} / N$ (they become isomorphic after base-change to $\mathbb{Z}[1/N, \zeta_N]$, which is \'etale over $\mathbb{Z}[1/N]$). At least, that would explain why Deligne and Rapoport only mention them in the chapter "Reduction modulo p". $\endgroup$
    – crocodile
    Commented Nov 20, 2012 at 11:45
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    $\begingroup$ @crocodile : I'm not familiar with the language of "stacks" used by Deligne-Rapoport but I think you're right. The isomorphism $w : \mathcal{A}_N \to \mathcal{B}_N$ in Deligne-Rapoport is none other than the Atkin-Lehner involution on $X_0(N)$ (or rather its model over $\mathbf{Z}[1/N]$). $\endgroup$
    – François Brunault
    Commented Nov 20, 2012 at 17:34
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There are in fact explicit equations (at least for the prime level) worked out in arXiv:math/0010272.

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