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Étale fundamental groups of non-Archimedean analytic spaces. (English) Zbl 0864.14012

In this paper, a space is either a rigid analytic space or the Berkovich space associated to this. The notion of étale covering map is introduced. This notion generalizes both finite étale coverings and topological coverings. On this notion the definition of the étale fundamental group is build. This group has a topology, is prodiscrete and maps to the topological and the algebraic fundamental group. For curves, i.e., spaces of dimension 1, some results on this étale fundamental group are given. The example of the étale covering map \(\log: \{z\in\mathbb{C}_p \mid|z-1 |<1\} \to\mathbb{C}_p\) seems to show that, even for the affine line, the étale fundamental group is too big to write down explicitly. Local \(\mathbb{Q}_l\)-systems are introduced and the connection with \(\mathbb{Q}_l\)-representations of the étale fundamental group is proven. Up to this point, the spaces were Berkovich spaces. The theory is then translated to the case of rigid analytic spaces. The motivation for developing the theory of étale fundamental groups comes from the work of Rapoport and Zink on moduli spaces for certain \(p\)-divisible groups and \(p\)-adic period maps. It is proved here that (a modification of) the period map is an étale covering map. Further results on the \(F\)-crystals and local systems, connected with the \(p\)-adic period map, are given. In particular, the \(p\)-adic period map provides an étale covering map of a projective space over, say, \(\mathbb{C}_p\). Thus the étale fundamental group of this projective space is not trivial!

MSC:

14G20 Local ground fields in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry
14E20 Coverings in algebraic geometry
14L05 Formal groups, \(p\)-divisible groups
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