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- Série 6 : Tome 27 (2018)

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- p. 199-264

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Nous étudions l’action du groupe de Galois arithmétique sur l’inertie géométrique attachée au groupe fondamental d’un (1-)champ algébrique, dans un contexte modéré. La situation est analogue mais aussi foncièrement distincte et plus complexe que celle, essentiellement bien comprise, qui concerne l’étude des groupes d’inertie procycliques associés aux composantes d’un diviseur à croisements normaux. Une grande partie du texte est consacrée à mettre en place les outils nécessaires à cette étude, elle-même en partie motivée par et appliquée à l’exemple important des champs de modules de courbes, dans lequel les groupes d’inertie en question correspondent aux automorphismes des courbes algébriques du type classifié par le champ.

We study the action of the arithmetic Galois group on the geometric inertia subgroups of the fundamental group, in a tame but typically stacky context. The problem is analogous but more involved than the by now fairly well-understood situation of the procyclic inertia subgroups associated with the components of a divisor with normal crossings. A significant part of the text is devoted to introducing the necessary tools for a study which is in part motivated by and applied to the important example of the moduli stacks of curves, where the geometric inertia groups correspond to the automorphisms of algebraic curves of the type classified by the stack.

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Pierre Lochak 1 ; Michel Vaquié 2

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[1] Dan Abramovich; Angelo Vistoli Compactifying the space of stable maps, J. Am. Math. Soc., Volume 15 (2002) no. 1, pp. 27-75 | DOI | MR | Zbl

[2] Francesc Bars Automorphism groups of genus 3 curves (Number Theory Seminar UAB-UB-UPC, Barcelona, January 2005, mat.uab.es/~francesc/mates/autgen3.pdf)

[3] José Bertin; Matthieu Romagny Champs de Hurwitz, Mém. Soc. Math. Fr., 125–126, Société Mathématique de France, 2011, 219 pages | Numdam | Zbl

[4] Nicolas Bourbaki Groupes et algèbres de Lie, Chapitre 5, Éléments de Mathematique, Masson, 1981 | Zbl

[5] Kenneth S. Brown Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1982, x+306 pages | Zbl

[6] Emilio Bujalance; Francisco Javier Cirre; José Manuel Gamboa; Grzegorz Gromadzki Symmetries of Compact Riemann Surfaces, Lecture Notes in Mathematics, 2007, Springer, 2010, xx+158 pages | MR | Zbl

[7] Fabrizio Catanese Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of Sing(𝔐 g ¯), Advances in geometric analysis (Advanced Lectures in Mathematics), Volume 21, Higher Education Press, 2012, pp. 281-306 | Zbl

[8] Benjamin Collas Action of a Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups of genus one, Int. J. Number Theory, Volume 8 (2012) no. 3, pp. 763-787 | DOI | MR | Zbl

[9] Benjamin Collas Action of the Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups in genus zero, J. Théor. Nombres Bordx., Volume 24 (2012) no. 3, pp. 605-655 | Numdam | DOI | MR | Zbl

[10] Benjamin Collas; Sylvain Maugeais On Galois action on stack inertia of moduli spaces of curves (2014) (https://arxiv.org/abs/1412.4644)

[11] Benjamin Collas; Sylvain Maugeais Composantes irréductibles de lieux spéciaux d ?espaces de modules de courbes, action galoisienne en genre quelconque, Ann. Inst. Fourier, Volume 65 (2015) no. 1, pp. 245-276 | Zbl | DOI

[12] Brian Conrad The Keel-Mori theorem via stacks (2005) (math.stanford.edu/~conrad/papers/coarsespace.pdf)

[13] Kevin Coombes; David Harbater Hurwitz families and arithmetic Galois groups, Duke Math. J., Volume 52 (1985), pp. 821-839 | MR | DOI | Zbl

[14] Maurizio Cornalba On the locus of curves with automorphisms, Ann. Mat. Pura Appl., Volume 149 (1987), pp. 135-151 | MR | DOI | Zbl

[15] Pierre Dèbes; Michel Emsalem On fields of moduli of curves, J. Algebra, Volume 211 (1999) no. 1, pp. 42-56 | MR | DOI | Zbl

[16] Pierre Deligne; David Mumford The irreducibility of the space of curves of given genus, Publ. Math., Inst. Hautes Étud. Sci., Volume 36 (1969), pp. 75-109 | DOI | Zbl | Numdam

[17] Allan L. Edmonds Surface symmetry I, Mich. Math. J., Volume 29 (1982), pp. 171-183 | MR | DOI | Zbl

[18] Hershel M. Farkas; Irwin Kra Riemann surfaces, Graduate Texts in Mathematics, 71, Springer, 1992, xvi+363 pages | Zbl

[19] G. González-Díez; William James Harvey Moduli of Riemann surfaces with symmetry, Discrete groups and geometry (London Mathematical Society Lecture Note Series), Volume 173, Cambridge University Press, 1992, pp. 75-93 | MR | DOI | Zbl

[20] Alexander Grothendieck Éléments de géométrie algébrique. I–IV, Publ. Math., Inst. Hautes Étud. Sci., Volume 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967) (Rédigé avec la collaboration de Jean Dieudonné) | Zbl

[21] Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz loceaux et globeaux (SGA 2) (Alexander Grothendieck, ed.), Advanced Studies in Pure Mathematics, 2, North-Holland, 1968, 287 pages | Zbl

[22] Revêtements Etales et Groupe Fondamental (SGA 1) (Alexander Grothendieck, ed.), Lecture Notes in Mathematics, 224, Springer, 1971, xxii+447 pages | Zbl

[23] Alexander Grothendieck; Jacob P. Murre The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, 208, Springer, 1971, viii+133 pages | MR | Zbl

[24] Seán Keel; Shigefumi Mori Quotients by groupoids, Ann. Math., Volume 145 (1997) no. 1, pp. 193-213 | MR | DOI | Zbl

[25] Gérard Laumon; Laurent Moret-Bailly Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 39, Springer, 2000, xii+208 pages | Zbl

[26] Pierre Lochak On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu, Volume 4 (2005) no. 3, pp. 443-508 | MR | DOI | Zbl

[27] Pierre Lochak Results and conjectures in profinite Teichmüller theory, Galois-Teichmüller theory and arithmetic geometry (Advanced Studies in Pure Mathematics), Volume 63, Mathematical Society of Japan, 2012, pp. 263-335 | DOI | Zbl

[28] Pierre Lochak; Hiroaki Nakamura; Leila Schneps Eigenloci of 5 points configurations on the Riemann sphere and the Grothendieck-Teichmüller group, Math. J. Okayama Univ., Volume 46 (2004), pp. 39-75 | Zbl

[29] Kay Magaard; Tanush Shaska; Sergey V. Shpectorov; Helmut Völklein The locus of curves with prescribed automorphism group, RIMS Kokyuroku, Volume 1267 (2002), pp. 112-141

[30] Makoto Matsumoto; Akio Tamagawa Mapping class group action versus Galois action on profinite fundamental groups, Am. J. Math., Volume 122 (2000) no. 5, pp. 1017-1026 | MR | DOI | Zbl

[31] Behrang Noohi Fundamental groups of algebraic stacks, M.I.T. (USA) (2000) (Ph. D. Thesis) | MR

[32] Behrang Noohi Fundamental groups of algebraic stacks, J. Inst. Math. Jussieu, Volume 3 (2004) no. 1, pp. 69-103 | MR | DOI | Zbl

[33] Fabrice Orgogozo Altérations et groupe fondamental premier à p, Bull. Soc. Math. Fr., Volume 131 (2003) no. 1, pp. 123-147 | MR | DOI | Zbl | Numdam

[34] Herbert Popp Stratifikation von Quotientenmannigfaltigkeiten und insbesondere der Modulmannigfaltigkeiten für Kurven, J. Reine Angew. Math., Volume 250 (1971), pp. 12-41 | MR | Zbl

[35] Matthieu Romagny Group actions on stacks and applications, Mich. Math. J., Volume 53 (2005) no. 1, pp. 209-236 | MR | DOI | Zbl

[36] Matthieu Romagny Composantes connexes et irréductibles en familles, Manuscr. Math., Volume 136 (2011) no. 1-2, pp. 1-32 | MR | DOI | Zbl

[37] Leila Schneps Special loci in moduli spaces of curves, Galois groups and fundamental groups (Mathematical Sciences Research Institute Publications), Volume 41, Cambridge University Press, 2003, pp. 217-275 | MR | Zbl

[38] Leila Schneps Automorphisms of curves and their role in Grothendieck-Teichmüller theory, Math. Nachr., Volume 279 (2006) no. 5-6, pp. 656-671 | Zbl

[39] Jean-Pierre Serre Corps locaux, Actualités Scientifiques et Industrielles, 1296, Hermann, 1962, 243 pages | Zbl

[40] William P. Thurston Geometry and topology of three-manifolds (1979) (http://library.msri.org/books/gt3m/)

[41] Angelo Vistoli Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., Volume 97 (1989) no. 3, 613.670 pages | MR | DOI | Zbl

[42] Vincent Zoonekynd The fundamental group of an algebraic stack (2001) (https://arxiv.org/abs/math/0111071)

[43] Vincent Zoonekynd Théorème de Van Kampen pour les champs algébriques, Ann. Math. Blaise Pascal, Volume 9 (2002) no. 1, pp. 101-145 | MR | DOI | Zbl | Numdam

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