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Curves and their fundamental groups [following Grothendieck, Tamagawa and Mochizuki]. (English) Zbl 0933.14015

Séminaire Bourbaki. Volume 1997/98. Exposés 835–849. Paris: Société Mathématique de France, Astérisque. 252, 131-150, Exp. No. 840 (1998).
This paper is a review of studies about the encoding of properties of an algebraic variety \(X\) in its algebraic fundamental group \(\pi _1(X\otimes _k \overline k)\), the profinite group which classifies finite étale coverings, defined by A. Grothendieck. The interesting case is when \(k\) is a proper subfield of \(\overline k \), since we can view the algebraic fundamental group as an extension \[ 0 \rightarrow \pi _1(X\otimes _k \overline k) \rightarrow \pi _1(X) \rightarrow \text{ Gal}(\overline k /k) \rightarrow 0. \] A result of S. Mochizuchi [see Invent. Math. 138, No. 2, 319-423 (1999)] shows that for hyperbolic curves \(X_1\), \(X_2\) over \(k\), where \(k\) is a finitely generated extension of \({\mathbb Q}_p\), any open map \( \pi _1(X_1) \rightarrow \pi _1(X_2)\), as \(\pi _1(X_2\otimes _k \overline k)\)-conjugacy class of map of extensions, is induced by a unique and dominant map \(X_1 \rightarrow X_2\). The main tool used here is \(p\)-adic Hodge theory.
For the entire collection see [Zbl 0911.00019].

MSC:

14H30 Coverings of curves, fundamental group
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
14E20 Coverings in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry
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