Questions about modular forms and the role of monodromy

Question

Let $N \geq 3$ and let $\Gamma=\Gamma_1(N)$, so that the moduli problem for elliptic curves with $\Gamma$-structure is fine. Let $Y=Y(\Gamma)$ be the corresponding moduli space.

In this context, one is usually given the definition of meromorphic forms and modular forms as follows. One has a universal elliptic curve $\mathcal{E}/Y$ and on it the sheaf of relative differentials $\Omega_{\mathcal{E}/Y}^1$, and $\omega$ is defined to be the pushforward of $\Omega_{\mathcal{E}/Y}^1$ along the projection $\pi:\mathcal{E}\rightarrow E$. Then $\omega$ is a line bundle on $Y$ which has the property that its fiber over a point $\tau$ is $H^0(E_{\tau},\Omega_{{E_\tau}}^1)$. One then defines meromorphic forms of weight k with level $\Gamma$ to be $H^0(Y,\omega^{\otimes{k}})$ and modular forms of weight k to be $H^0(X,\omega^{\otimes{k}})$ for a certain extension of $\omega$ from $Y$ to $X$.

Now it seems to me the whole reason this definition should give the correct notion of modular forms is that there is a monodromy action of $\Gamma$ on a differential $dz$ on $E_{\tau}$, namely, $dz$ should be mapped to $(c\tau+d)dz$ when we go along a curve connecting $\tau$ and $\gamma(\tau)$ in the upper half plane. However strictly speaking there seems to be no local system appearing in the above definition, and therefore the role of monodromy here is unclear to me.

With this in mind, here is an alternate definition which may or may not make sense. Let $\Omega_{\mathbb{C}}^1=\mathbb{C}dz$ be the $\mathbb{C}$ vector space of dimension $1$ generated by the symbol $dz$, and let $M(\mathbb{H})$ be meromorphic functions on upper half plane. Then we have a homomorphism $$\pi_1(Y)=\Gamma\rightarrow Aut_{M(\mathbb{H})}(\Omega_{\mathbb{C}}^1 \otimes_{\mathbb{C}} M(\mathbb{H})),$$with $\gamma$ mapping $dz \otimes f(\tau)$ to $dz \otimes (c\tau+d)f(\gamma^{-1}(\tau))$. This is a local system, and hence gives rise to a locally constant sheaf $\tilde{\omega}$ on $Y$, which is locally $\Omega_{\mathbb{C}}^1 \otimes_{\mathbb{C}} M(\mathbb{H})$.

The point is that for each $\tau$ we have a natural evaluation map $$\Omega_{\mathbb{C}}^1 \otimes_{\mathbb{C}} M(\mathbb{H})\rightarrow H^0(E_{\tau},\Omega_{{E_\tau}}^1)$$ which specialises the action of $\Gamma$ on $\Omega_{\mathbb{C}}^1 \otimes_{\mathbb{C}} M(\mathbb{H})$ to the usual one on $H^0(E_{\tau},\Omega_{{E_\tau}}^1)$.

Then it seems to me that sections of $\tilde{\omega}^{\otimes k}$ invariant under the action of $\Gamma$ should be exactly the same as modular forms. Thus, my first question is:

Question 1. Let $(\tilde{\omega}^{\otimes k})^{\Gamma}$ be these sections invariant under $\Gamma$. Is it true that $(\tilde{\omega}^{\otimes k})^{\Gamma}=\omega^{\otimes k}$?

At the very least it seems like they both have the same fibers over each point $\tau$.

Now whether this is true or not, this $\tilde{\omega}$ seems to be defined only over $\mathbb{C}$, unlike $\omega$ which is always defined. Which makes me wonder:

Question 2. If the answer to question 2 is yes, should it be possible to construct $\tilde{\omega}$ over a more general base than $\mathbb{C}$?

Thanks!

Answer 1

Monodromy does not appear in such a nice way because $\omega$ is a coherent sheaf, not a locally constant sheaf. Thus there is not necessarily a way of continuing a local section along a path in $\mathbb H$.

One way to connect this to monodromy is to use the Eichler-Shimura isomorphism, which gives an isomorphism between modular forms of weight $k$ and $\operatorname{Sym}^{k-2}$ of a natural rank $2$ local system on $Y$. This definition can be made to work over a general field using $\ell$-adic cohomology.

Your construction seems overly complicated. One can define on action of $\Gamma$ on $H^0( \mathbb H, \tilde{\omega}^{\otimes k} )$ for which meromorphic forms are the $\Gamma$-invariants. If desired, one can view $H^0( \mathbb H, \tilde{\omega}^{\otimes k} )$ as a local system on $Y$ using this $\Gamma$-action. There is no reason to introduce meromorphic functions on $\mathbb H$ or tensor products over $\mathbb C$ of two different spaces of functions (which seem terrible to me).