I have never understood the definition of a modular form in terms of differential forms. What is formally going on when we rearrange the equation $\frac{gz}{f(z)} = (\frac{d(gz)}{dz})^{-k}$ to the equation $f(gz)d(gz)^k = f(z)dz^k$?
What the formal definition of a "differential form of weight $k$?" What is meant by $f(z)dz^k$ where $k$ is an integer and $f$ is a meromorphic function on the upper half plane?
I know the basics of differential forms on smooth manifolds. If $M$ is a real smooth manifold, then a differential $k$-form on $M$ is a smooth section of the map $\Lambda^k (T^{\ast} M) \rightarrow M$, where $T^{\ast} M$ is the contangent bundle of $M$ and $\Lambda^k$ denotes exterior power.
The attached excerpt is from Serre, A Course in Arithmetic.
