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It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some local studies of Shimura varieties), or the Tate uniformizaation of abelian varieties with multiplicative reduction (which we roughly think of as uniformized by tori). I would like to know what other kinds of spaces have been used in $p$-adic uniformization other than these two cases, as I haven't yet found much in the literature.

And in general, what kind of spaces could one expect to appear in the $p$-adic uniformization as the "universal covering" spaces? what is the $p$-adic counterpart of "simply connected" spaces? Since smooth Berkovich analytic spaces are locally contractible, can we expect such spaces exist in the sense of Berkovich?

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Simply connected certainly makes sense in Berkovich theory and locally contractible spaces admit universal coverings that are simply connected. In this direction, Hrushovski and Loeser also proved that Berkovich analytifications of algebraic varieties are locally contractible (see arXiv:1009.0252). –  Jérôme Poineau Jan 31 '12 at 7:29
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There is another type of uniformization introduced in Mochizuki's book Foundations of $p$-adic Teichmüller theory. It uses curves equipped with nilpotent indigenous bundles.

I don't see what local contractibility has to do with non-existence of simply connected spaces. The finite étale covers of the affine line and the punctured affine line are very different, even though the underlying Berkovich spaces look almost the same, and are both contractible in the topological sense.

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