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?