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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.

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    $\begingroup$ You're mixing up two unrelated $q$'s: really write $\mathbf{C}[\![q]\!] ({\rm{d}}t/t)^{\otimes k}$ for a specific relative 1-form "${\rm{d}}t/t$". In brief: prove the relevant Tate family over the punctured open unit disc has one (up to unique isomorphism) extension to an analytic generalized elliptic curve with smooth total space and special fiber with the "expected" number of analytic irreducible components, so the analytification of the abstract algebraic universal family pulls back to this explicit analytic model, and compare on infinitesimal special fibers via formal completions along 1. $\endgroup$
    – nfdc23
    Apr 18, 2016 at 0:29
  • $\begingroup$ Thank you for your comment. I'm afraid I haven't the necessary background to check what I think you are suggesting. Do you know of any reference where I can read about it? I have found these great notes of Brian Conrad math.stanford.edu/~conrad/248BPage/handouts/qexp.pdf where he writes that he checked the compatibility in class. Is there anyone out there that is lucky to possess some notes of that class? $\endgroup$
    – Bear
    Apr 18, 2016 at 11:32
  • $\begingroup$ It seems from what is written in the link that the argument given in that class is the same one I have in mind, a curious coincidence. I don't have a reference, since I came up with an argument when I was learning these things myself and was frustrated by not being able to find a reference. The books/papers of Kai-Wen Lan should address the issue, albeit in a much more general framework. $\endgroup$
    – nfdc23
    Apr 18, 2016 at 15:31

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