I would like to check that algebraic and analytic q-expansion of a modular form coincide.
I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular curve over $\text{Spec}\mathbb{C}$ and $X^\text{an}$ its analytification, there is a natural map of ringed spaces $\iota:X^\text{an}\to X$ which, for any point $x\in X^\text{an}$, gives a natural isomorphism $\hat{\cal{O}}_{X,x}\overset{\sim}{\to}\cal{O}_{X^\text{an},x}$ between the completion of the local ring of the scheme $X$ at $x$ and the local ring of the Riemann surface $X^\text{an}$ at $x$. Since our modular curve $X$ is proper, by GAGA we have an isomorphism $H^0(X, \underline{\omega}^{\otimes k})\cong H^0(X^\text{an}, \underline{\omega}_\text{an}^{\otimes k})$ and so we have an equivalence between Katz's definition of modular forms and the classical one.
If $\infty\in X$ is a cusp, by interpreting the Tate curve as universal deformation of the Neron polygon, we get a canonical isomorphism $\mathbb{C}[[q]]\cong\hat{\cal{O}}_{X,\infty}$, and using the canonical differential $dt/t$ on the Tate curve we get a canonical isomorphism
$\mathbb{C}[[q]](\frac{dt}{t})^{\otimes k}\cong\big(\underline{\omega}_\infty^{\otimes k}\big)^\wedge\cong\underline{\omega}_{\text{an},\infty}^{\otimes k}$.
How can one proceed from here?
Edited following nfdc23's comment.