I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", <http://arxiv.org/pdf/math/0311202v1.pdf>, and I have a question regarding the nature of the function field for $X0(N)$: if I have a modular function for $Γ\_0(N)$ that has a rational $q$-expansion at $\infty$, does it follow that $f∈\mathbf{Q}(j,j\_N)$?


Allow to me to elaborate on some discoveries I've made reading the above paper, and playing with sage:


I understand that the function field over $\mathbf{C}$ for $X\_0(N)$ is $\mathbf{C}(j,j\_N)$, and that the $\mathbf{Q}$-function field given by $\mathbf{Q}(j,j\_N)$ defines the curve $X\_0(N)$ over the rationals. Thus, any modular function for $\Gamma\_0(N)$ can be written as rational function in $j$ and $j\_N$.


Now, for the case of level $N=1$ modular functions, it is stated in the paper that if $f$ is a modular function that has a fourier expansion at $\infty$ with rational coefficients, then $f$ in fact lies in $\mathbf{Q}(j)$, which seems intuitive. However, I have recently discovered that the analogous fact does not hold for modular functions for $\Gamma\_0(N)$ that have a rational fourier expansion at $\infty$. Using a paper of Maier, I have been able to play with the hauptmoduln for $\Gamma\_0(N)$ when the group has genus zero. These functions have rational $q$-expansions, but don't necessarily lie in $\mathbf{Q}(j,j\_N)$. Indeed, using the function $t\_2$ in his paper (a hauptmodul for $\Gamma\_0(2)$ that has rational $q$-expansion), I computed that
$$t\_2(1/4\sqrt{-7}−1/4)$$
is an algebraic integer of degree $2$. In particular, it is definitely not a rational number. On the other hand, the theory of complex multiplication ensures that both $j(1/4\sqrt{-7}−1/4)$ and $j\_2(1/4\sqrt{-7}−1/4)$ are integers. Therefore, $t\_2$ cannot lie in $\mathbf{Q}(j,j2)$, since otherwise $t\_2(1/4\sqrt{-7}−1/4)$ would have to be rational as well.


What is troubling about this fact is that, in the paper of Cox et al., it is stated that if $f$ is a modular function for $\Gamma\_0(N)$ and if $\tau$ is an elliptic point of order 2 for $\Gamma\_0(N)^\dagger$, but not one for $\Gamma\_0(N)$, then $f(\tau)$ must lie in the maximal abelian extension of $K=\mathbf{Q}(\tau)$ and that this follows from basic complex multiplication theory. Now, if it were the case that $f\in\mathbf{Q}(j,j\_N)$, then I'd be happy to believe this, but I've just seen that this is not the case. Therefore, I don't see how basic complex multiplication theory says anything about the nature of $f(\tau)$ in this case.


Lastly, as mentioned above, it is stated explicitly that for level $N=1$, we do know that such a modular function with rational $q$-expansion must lie in $\mathbf{Q}(j)$. Does this result remain true for $X\_0(N)$ when it has genus zero? That is, after fixing a hauptmodul $h$ with rational fourier expansion, so that any modular function for $\Gamma\_0(N)$ can be written as a rational function in $h$, is it true that those functions with rational $q$-expansions can be written as a rational function of $h$ with coefficients in $\mathbf{Q}$?


If anyone can shed some led on any part of this problem, I would very much appreciate it!



Modular Functions with Rational Fourier Expansions
The statement about being in ${\bf Q}(j,j\_N)$ if the $q$-expansion is rational is true. See Prop 12.7 (a) in Cox's book "Primes of the form $x^2 + Ny^2$". Moreover, part (b) of that proposition tells you that you can specialize when the partial derivative $\partial \Phi\_N(x,y)/ \partial(x)$ does not vanish at that point. That's not the case for the CM point you've chosen. In general if you have a genus zero curve for $X\_0(N)$ and $h$ a hauptmodul, then this $q$-expansion fact will work, and you can certainly specialize as long as the denominator of the rational function in $h$ doesn't vanish.


