Consider a torsion-free congruence subgroup $\Gamma\le SL_2(\mathbb{Z})$ (for example, $\Gamma(N)$ for $N\ge 3$, or $\Gamma_1(N)$ for $N\ge 4$).

By a meromorphic modular form for $\Gamma$ of weight $k$, I mean a holomorphic function $f$ on $\mathcal{H}$ with $f(\gamma z) = (cz+d)^kf(z)$ for $\gamma\in\Gamma$, which is meromorphic at the cusps.

Let $M_k(\Gamma)$ be the $\mathbb{C}$-vector space of meromorphic modular forms for $\Gamma$ of weight $k$. Is the graded ring $\bigoplus_{k\ge 0} M_k(\Gamma)$ generated in degree 1?

I believe this follows from the geometric interpretation of modular forms as sections of line bundles on moduli stacks (at least when the moduli stack is representable), but I just wanted to check that this is also known from an analytic perspective as well.

Also, if the above is true, then can someone explain roughly why this fails if $\Gamma$ is not torsion-free? (ie, are line bundles on stacks, viewed as the Spec of the symmetric algebra of the corresponding invertible module *not* necessarily generated in degree 1? Perhaps I don't have the right definition of a line bundle on a stack)

holomorphicmodular forms: see math.stackexchange.com/questions/96395 (for $\Gamma$ sufficiently small) and mathoverflow.net/questions/66819 (for general $\Gamma$). This is a slightly different question, of course, but the methods might help for your question too. $\endgroup$ – David Loeffler Sep 25 '16 at 8:21