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Jonah's question makes me wonder: What is with uniformization in algebraic/arithmetic geometry? E.g. this article by Faltings seems to be about that, the Shimura-Taniyama statement too, Mochizuki discusses a p-adic version of Fuchsian uniformization of hyperbolic curves. Do you know surveys or expositions of the theme? Or is 'uniformization' a mistaken concept in the context of arithmetic geometry (a remark in Faltings article sounds as if saying that)?

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Presumably, you want to look at uniformizations of curves, since blow-ups make it difficult to classify covers of higher dimensional varieties. Scheme-theoretically, there doesn't seem to be a good notion of uniformization, but one can get good arithmetic results using analytification.

You already mentioned a couple references. For the complex-analytic view, I think any book on modular forms (e.g., Shimura) should have some treatment. Mochizuki's work in the nonarchimedean setting has a counterpart in the maximally degenerate case of Mumford curves. There is a book by Gekeler and van der Put on uniformizations of these (essentially using the Bruhat-Tits building for SL2). I think Darmon and Dasgupta have done some arithmetic work using this "upper half plane" uniformization. It historically originates from Tate's work around 1960 on rigid analytic spaces (apparently, Grothendieck expressed doubt that such a theory would exist).

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