# Arguing that weakly holomorphic modular forms give rise to Katz modular forms

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.

• Working in the purely analytic setting as you are, how are you inferring meromorphicity rather than the possibility of essential singularities? Saying "evaluate at the Tate curve" doesn't make the issue of essential singularities disappear. Also, since there is Chow's Lemma for separated Artin stacks of finite type, to prove a GAGA-type theorem (you only need DM stacks, which is technically simpler) you can bootstrap from the cases of proper algebraic spaces and/or schemes exactly as Grothendieck does from the projective to the proper case for schemes in Expose XII of SGA1. – nfdc23 Oct 30 '17 at 5:31
• @nfdc23 If $F$ is a Katz modular form, which determines a classical modular form $f$, then evaluating at the Tate curve over $\mathbb{C}((q^{1/n}))$ gives you an element of $\mathbb{C}((q^{1/n}))$ which I believe is precisely the $q$-expansion of $f$ at some cusp, depending on the level structure one chooses - but elements of $\mathbb{C}((q^{1/n}))$ are laurent series, and all level structures are defined over $\mathbb{C}((q^{1/n}))$ so doesn't that mean $f$ can't have any essential singularities at the cusps? – stupid_question_bot Oct 30 '17 at 5:40
• OK, I didn't read your entire question (it was too long for me), so I was just guessing that you were starting from the purely analytic side, which I now see you weren't doing. Being "weakly holomorphic" with pole-order below some bound at the cusps corresponds to a global section of the evident line bundle on the proper stack by imposing some twist of the usual line bundle by a positive power of the inverse ideal sheaf of the cusps, so you can just applying GAGA to conclude without having to say anything about "Tate curves" or $q$-expansions. – nfdc23 Oct 30 '17 at 6:03
• @nfdc23 Right, so my questions are: (1) Where can I find a statement of GAGA for Deligne-Mumford stacks (which ideally doesn't use infinity-stack terminology), and (2) Given a classical modular form $f$, meromorphic at all cusps, how does one define a section of the relevant line bundle on the proper stack $[\overline{\mathcal{H}}/\Gamma]$? (I think I can do it on the open substack $[\mathcal{H}/\Gamma]$, but I'm still not sure how to do it near the cusps. – stupid_question_bot Oct 30 '17 at 22:21
• In my first comment I sketched how to prove GAGA for DM stacks; I don't know a reference. A classical form meromorphic at the cusps is a global section of a twist by a positive power of the inverse ideal sheaf of the cuspidal substack against the sheaf without allowing poles, and the latter is built as in the algebro-geometric case from the universal generalized elliptic curve over the moduli stack. You can define "analytifications" by descent from higher-level scheme cases; the notation $[\overline{\mathcal{H}}/\Gamma]$ makes no sense since $\overline{\mathcal{H}}$ isn't a complex manifold. – nfdc23 Oct 31 '17 at 14:50