Let $\Gamma = \Gamma_1(n)\le\text{SL}_2(\mathbb{Z})$ for some $n$ (might even take $n = 1$).

Let $\mathcal{H}$ be the upper half plane, then on $\mathcal{H}$ we have a line bundle $\mathcal{H}\times\mathbb{C}$. Let $\omega_\Gamma^{an}$ denote the set of isomorphism classes of pairs $(E,P,\omega)$, where $E$ is an elliptic curve over $\mathbb{C}$ and $\omega$ a nonzero holomorphic differential on $E$, and $P$ a point of order $n$.

We may define a map $\mathcal{H}\times\mathbb{C}^\times\rightarrow \omega_\Gamma$ sending $(\tau,t)\mapsto (E_\tau,P_\tau,dz)$ where $E_\tau := \mathbb{C}/\langle 1,\tau\rangle$, $P_\tau := \frac{1}{n} + \langle 1,\tau\rangle$, and $dz$ is a fixed differential on $\mathbb{C}$ pushed forward onto $E_\tau$.

The map $\mathcal{H}\times\mathbb{C}^\times\rightarrow\omega_\Gamma^{an}$ above induces an isomorphism $$(\mathcal{H}\times\mathbb{C}^\times)/\Gamma\cong\omega_\Gamma^{an}$$ where $\Gamma$ acts on $\mathcal{H}\times\mathbb{C}^\times$ via the formula: $$\gamma := \begin{bmatrix} a & b \\ c & d \end{bmatrix}\qquad\gamma(\tau,t) := (\frac{a\tau + b}{c\tau + d},(c\tau + d)t)$$

Forgetting the differential $\omega$ yields a map to the analytic stack $$\omega_\Gamma^{an}\rightarrow[\mathcal{H}/\Gamma]$$ which almost realizes $\omega_\Gamma^{an}$ as a line bundle on $[\mathcal{H}/\Gamma]$, except not quite since the fibers of this map are $\mathbb{C}^\times$. Instead, $\omega_\Gamma^{an}$ sits as an open subspace of the analytic moduli stack $\overline{\omega_\Gamma^{an}}$ of elliptic curves equipped with a differential (possibly zero) and a point of order $n$. Then, the stack $\overline{\omega_\Gamma^{an}}$ should be a line bundle over $[\mathcal{H}/\Gamma]$.

Let $|\overline{\omega_{\Gamma}^{an}}|$ denote the coarse moduli space, then $$|\overline{\omega_{\Gamma}^{an}}| = (\mathcal{H}\times\mathbb{C})/\Gamma$$ and this space admits a natural complex structure given by the projection $\mathcal{H}\times\mathbb{C}\rightarrow(\mathcal{H}\times\mathbb{C})/\Gamma$. There is a standard argument that any holomorphic function $f : \mathcal{H}\rightarrow\mathbb{C}$ satisfying $f(\gamma \tau) = (c\tau + d)^kf(\tau)$ gives rise to a unique holomorphic function $F : |\overline{\omega_\Gamma^{an}}|\rightarrow\mathbb{C}$ satisfying $$F(E,P,\lambda\omega) = \lambda^{-k}F(E,P,\omega),\quad\text{and}\quad F(E_\tau,P_\tau,dz) = f(\tau)$$ In turn, any such a function $F$ can be used to define a section $F' : [\mathcal{H}/\Gamma]\rightarrow (\omega_\Gamma^{an})^{\otimes k}$ by: $$F'(E_\tau,P_\tau) = (E_\tau,P_\tau,F(E_\tau,P_\tau,\omega)\omega^{\otimes k})\qquad \text{$\omega\ne 0$}$$ where the weight-$k$ homogeneity of $F$ means that $F(E_\tau,P_\tau,\omega)\omega^{\otimes k}$ does not depend on the choice of $\omega\ne 0$.

Conversely, given $F'$, one can define $F : |\overline{\omega_\Gamma^{an}}|\rightarrow\mathbb{C}$ via $$F(E_\tau,P_\tau,\omega) = \left\{\begin{array}{ll}F'(E_\tau,P_\tau)/\omega^{\otimes k} & \omega\ne 0 \\ 0 & \omega = 0\end{array}\right.$$ and given $F$, one can recover $f$ by restricting to triples of the form $(E_\tau,P_\tau,dz)$.

Now, algebraically, let $\overline{\omega_\Gamma^{alg}}$ denote the moduli stack (over $\mathbb{C})$ of elliptic curves over $\mathbb{C}$-schemes equipped with a possibly zero differential, and let $\mathcal{M}(\Gamma)$ be the moduli stack over $\mathbb{C}$ of elliptic curves with $\Gamma$-structures.

Thus, $(\overline{\omega_\Gamma^{alg}})^{\otimes k}\rightarrow\mathcal{M}(\Gamma)$ is the algebraic version of $(\overline{\omega_\Gamma^{an}})^{\otimes k}\rightarrow [\mathcal{H}/\Gamma]$. The sections of the former are often called Katz modular forms (over $\mathbb{C}$) of weight $k$ for $\Gamma$, and for now I think I understand how Katz modular forms give rise to sections of the latter, which in turn give rise to holomorphic functions $f : \mathcal{H}\rightarrow\mathbb{C}$ which are weight $k$-invariant under $\Gamma$. By evaluating at the Tate curve, I believe Katz modular forms give rise to weakly holomorphic modular forms. Ie, $f$ is moreover meromorphic at all cusps.

My main question is:

Does every weight $k$ weakly holomorphic modular form $f : \mathcal{H}\rightarrow\mathbb{C}$ (for $\Gamma$), or equivalently a section of $(\overline{\omega_\Gamma^{an}})^{\otimes k}\rightarrow[\mathcal{H}/\Gamma]$ give rise to a Katz modular form? (ie, a section of $(\overline{\omega_\Gamma^{alg}})^{\otimes k}\rightarrow\mathcal{M}(\Gamma))$

If not, at least do the holomorphic modular forms give rise to Katz modular forms? How would we argue this?

Let $\Gamma = \Gamma_1(n)\le\text{SL}_2(\mathbb{Z})$ for some $n$ (I don't want to assume that $\Gamma$ is torsion-free)

Let $\mathcal{H}$ be the upper half plane, then on $\mathcal{H}$ we have a line bundle $\mathcal{H}\times\mathbb{C}$. Let $\omega_\Gamma^{an}$ denote the set of isomorphism classes of pairs $(E,P,\omega)$, where $E$ is an elliptic curve over $\mathbb{C}$ and $\omega$ a nonzero holomorphic differential on $E$, and $P$ a point of order $n$.

We may define a map $\mathcal{H}\times\mathbb{C}^\times\rightarrow \omega_\Gamma^{an}$ sending $(\tau,t)\mapsto (E_\tau,P_\tau,dz)$ where $E_\tau := \mathbb{C}/\langle 1,\tau\rangle$, $P_\tau := \frac{1}{n} + \langle 1,\tau\rangle$, and $dz$ is a fixed differential on $\mathbb{C}$ pushed forward onto $E_\tau$.

The map $\mathcal{H}\times\mathbb{C}^\times\rightarrow\omega_\Gamma^{an}$ above induces an isomorphism $$(\mathcal{H}\times\mathbb{C}^\times)/\Gamma\cong\omega_\Gamma^{an}$$ where $\Gamma$ acts on $\mathcal{H}\times\mathbb{C}^\times$ via the formula: $$\gamma := \begin{bmatrix} a & b \\ c & d \end{bmatrix}\qquad\gamma(\tau,t) := (\frac{a\tau + b}{c\tau + d},(c\tau + d)t)$$

Forgetting the differential $\omega$ yields a map to the analytic stack $$\omega_\Gamma^{an}\rightarrow[\mathcal{H}/\Gamma]$$ which almost realizes $\omega_\Gamma^{an}$ as a line bundle on $[\mathcal{H}/\Gamma]$, except not quite since the fibers of this map are $\mathbb{C}^\times$. Instead, $\omega_\Gamma^{an}$ sits as an open subspace of the analytic moduli stack $\overline{\omega_\Gamma^{an}}$ of elliptic curves equipped with a differential (possibly zero) and a point of order $n$. Then, the stack $\overline{\omega_\Gamma^{an}}$ should be a line bundle over $[\mathcal{H}/\Gamma]$.

Let $|\overline{\omega_{\Gamma}^{an}}|$ denote the coarse moduli space, then $$|\overline{\omega_{\Gamma}^{an}}| = (\mathcal{H}\times\mathbb{C})/\Gamma$$ and this space admits a natural complex structure given by the projection $\mathcal{H}\times\mathbb{C}\rightarrow(\mathcal{H}\times\mathbb{C})/\Gamma$. There is a standard argument that any holomorphic function $f : \mathcal{H}\rightarrow\mathbb{C}$ satisfying $f(\gamma \tau) = (c\tau + d)^kf(\tau)$ gives rise to a unique holomorphic function $F : |\overline{\omega_\Gamma^{an}}|\rightarrow\mathbb{C}$ satisfying $$F(E,P,\lambda\omega) = \lambda^{-k}F(E,P,\omega),\quad\text{and}\quad F(E_\tau,P_\tau,dz) = f(\tau)$$ In turn, any such a function $F$ can be used to define a section $F' : [\mathcal{H}/\Gamma]\rightarrow (\omega_\Gamma^{an})^{\otimes k}$ by: $$F'(E_\tau,P_\tau) = (E_\tau,P_\tau,F(E_\tau,P_\tau,\omega)\omega^{\otimes k})\qquad \text{$\omega\ne 0$}$$ where the weight-$k$ homogeneity of $F$ means that $F(E_\tau,P_\tau,\omega)\omega^{\otimes k}$ does not depend on the choice of $\omega\ne 0$.

Conversely, given $F'$, one can define $F : |\overline{\omega_\Gamma^{an}}|\rightarrow\mathbb{C}$ via $$F(E_\tau,P_\tau,\omega) = \left\{\begin{array}{ll}F'(E_\tau,P_\tau)/\omega^{\otimes k} & \omega\ne 0 \\ 0 & \omega = 0\end{array}\right.$$ and given $F$, one can recover $f$ by restricting to triples of the form $(E_\tau,P_\tau,dz)$.

Now, algebraically, let $\overline{\omega_\Gamma^{alg}}$ denote the moduli stack (over $\mathbb{C})$ of elliptic curves over $\mathbb{C}$-schemes equipped with a possibly zero differential, and let $\mathcal{M}(\Gamma)$ be the moduli stack over $\mathbb{C}$ of elliptic curves with $\Gamma$-structures.

Thus, $(\overline{\omega_\Gamma^{alg}})^{\otimes k}\rightarrow\mathcal{M}(\Gamma)$ is the algebraic version of $(\overline{\omega_\Gamma^{an}})^{\otimes k}\rightarrow [\mathcal{H}/\Gamma]$. The sections of the former are often called Katz modular forms (over $\mathbb{C}$) of weight $k$ for $\Gamma$, and for now I think I understand how Katz modular forms give rise to sections of the latter, which in turn give rise to holomorphic functions $f : \mathcal{H}\rightarrow\mathbb{C}$ which are weight $k$-invariant under $\Gamma$. By evaluating at the Tate curve, I believe Katz modular forms give rise to weakly holomorphic modular forms. Ie, $f$ is moreover meromorphic at all cusps.

My main question is:

Does every weight $k$ weakly holomorphic modular form $f : \mathcal{H}\rightarrow\mathbb{C}$ for $\Gamma$, via the corresponding section of $(\overline{\omega_\Gamma^{an}})^{\otimes k}\rightarrow[\mathcal{H}/\Gamma]$, give rise to a Katz modular form? (ie, a section of $(\overline{\omega_\Gamma^{alg}})^{\otimes k}\rightarrow\mathcal{M}(\Gamma))$

If not, at least do the holomorphic modular forms give rise to Katz modular forms? How would we argue this? Essentially, this is a question about GAGA for stacks - for which I'm having difficulty finding a good reference, so references would also be appreciated!

(A related question is tantalizingly "answered" in Brian Conrad's notes, but unfortunately the explanation was apparently given in class and not discussed in the notes)

Secondly, in the definition of the section $F'$, I cheated a bit since I only defined it on "points", mostly because I'm not sure what the correct definition would be. If someone could point out the correct definition, that would also be greatly appreciated.