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A note on p-adic uniformization. (English) Zbl 0624.32018
A connected analytic space over a nonarchimedean field K is called simply connected if it has no nontrivial connected analytic coverings. The author proves the following two criteria for a quasi-compact K-analytic space X to be simply connected: (i) there is a morphism with simply connected fibres from X to a simply connected analytic space; (ii) X has irreducible and smooth reduction.
These criteria are applied to show that in all known examples of p-adic uniformization (Mumford curves, certain abelian varieties, Mumford surfaces) the uniformization is indeed by the universal covering.
The proofs rely on the author’s previous paper [Compos. Math. 45, 165-198 (1982; Zbl 0491.14014)]. (Note that in Lemma 4.1 the additional hypothesis $$\dim Z\geq 2''$$ has to be inserted.)
Reviewer: F.Herrlich

##### MSC:
 32P05 Non-Archimedean analysis 14G20 Local ground fields in algebraic geometry 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14E20 Coverings in algebraic geometry