Courbes semi-stables et groupe fondamental en géométrie algébrique: Luminy, Decembre 1998

This volume contains detailed expositions of talks given during an instructional conference held at Luminy in December 1998, which was devoted to classical and recent results concerning the fundamental group of algebraic curves, especially over finite and local fields. The scientific guidance of the conference was supplied by M. Raynaud, a leading expert in the field. The purpose of this volume is twofold. Firstly, it gives an account of basic results concerning rigid geometry, stable curves, and algebraic fundamental groups, in a form which should make them largely accessible to graduate students mastering a basic course in modern algebraic geometry. However classic, most of this material has not appeared in book form yet. In particular, the semi-stable reduction theorem for curves is covered with special care, including various detailed proofs. Secondly, it presents self-contained expositions of important recent developments, including the work of Tamagawa on Grothendieck's anabelian conjecture for curves over finite fields, and the solution by Raynaud and Harbater of Abhyankar's conjecture about coverings of affine curves in positive characteristic. These expositions should be accessible to research students who have read the previous chapters. They are also aimed at experts in number theory and algebraic geometry who want to read a streamlined account of these recent advances. Table des matiere Avant-propos Chapitre 1 Géométrie rigide et géométrie formelle Marco A. Garuti Chapitre 2 Disques et Couronnes ultramétriques Yannick Henrio Chapitre 3 The Picard Functor Urs T. Hartl Chapitre 4 Module des courbes stables David Mauger Chapitre 5 Réduction semi-stable des courbes d'après Artin, Deligne, Grothendieck, Mumford, Saito, Winters, ... Ahmed Abbes Chapitre 6 Mumford-Tate curves Thorsten Schmechta Chapitre 7 U niformization of Shimura curves by the p-adic upper half plane Klaus Kiinnemann Chapitre 8 Le théorème de Gabber-Lütkebohmert François Loeser Chapitre 9 Fundamental group Ariane Mezard Chapitre 10 Construction of Covers with Formal and Rigid Geometry Rachel J. Pries Chapitre 11 Le théorème de spécialisation du groupe fondamental Fabrice Orgogozo & Isabelle Vidal Chapitre 12 Le théorème de Tamagawa 1 Tamâs Szamuely Chapitre 13 Le théorème de Tamagawa II David Harari Chapitre 14 Le groupe fondamental sauvage d'une courbe affine en caractéristique p > 0 Philippe Gille Chapitre 15 La conjecture d'Abhyankar 1 : construction de revêtements en géométrie rigide Antoine Chambert-Loir Chapitre 16 Abhyankar's Conjecture II : the use of semi-stable curves Mohamed Sa!di Chapitre 17 The p-rank of curves and covers of curves Irene I. Bouw Chapitre 18 Theta divisors and the Frobenius morphism David A. Madore Ce livre rassemble les textes des conférences donnees au CIRM à Luminy du 30 novembre 1998 au 4 décembre, avec pour thème : Le groupe fondamental des courbes en géométrie algébrique. Jean-benoît Bost, François Loeser, Michel Raynaud, éditeurs. Includes Bibliographical References.