Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque
Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 245-276.

Dans cet article, nous caractérisons l’action du groupe de Galois absolu sur les groupes d’inertie champêtre géométriques cycliques et sans factorisation étale du groupe fondamental géométrique des espaces de modules de courbes marquées. Nous établissons par ailleurs la même action sur les éléments de torsion profinis d’ordre premier en genre 2.

In this paper we characterise the action of the absolute Galois group on the geometric finite cyclic groups without étale factorization of stack inertia of the profinite geometric fundamental group of moduli spaces of marked curves. As a complementary result, we give the same action on prime order profinite elements in genus 2.

DOI : https://doi.org/10.5802/aif.2930
Classification : 11R32,  14H10,  14H30,  14H45
Mots clés : groupe fondamental algébrique, inertie champêtre, lieu spécial, groupes bons
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Collas, Benjamin; Maugeais, Sylvain. Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque. Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 245-276. doi : 10.5802/aif.2930. http://archive.numdam.org/articles/10.5802/aif.2930/

[1] Bertin, J.; Romagny, M. Champs de Hurwitz, Mémoire de la SMF, 125-126, SMF, 2011 (arXiv :math/0701680v1) | Numdam | MR 2920693 | Zbl 1242.14025

[2] Broughton, S. A. The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl., Volume 37 (1990) no. 2, pp. 101-113 | Article | MR 1080344 | Zbl 0747.32017

[3] Catanese, F. 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, Volume 21, Int. Press, Somerville, MA, 2012, p. 281-306, arXiv :1011.0316v1 | MR 3077261

[4] Collas, B. Action of a Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups of genus one, International Journal of Number Theory, Volume 84 (2012) no. 3, pp. 763-787 | Article | MR 2904929 | Zbl 1288.14015

[5] Collas, B. Action of the Grothendieck-Teichmüller group on torsion elements of mapping class groups in genus zero, Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 605-622 | Article | EuDML 251074 | Numdam | MR 3010631 | Zbl 1278.14040

[6] Cornalba, M. On the locus of curves with automorphisms, Ann. Mat. Pura Appl. (4), Volume 149 (1987), pp. 135-151 | Article | MR 932781 | Zbl 0649.14013

[7] Cornalba, M. Erratum : “On the locus of curves with automorphisms” [Ann. Mat. Pura Appl. (4) 149 (1987), 135–151], Ann. Mat. Pura Appl. (4), Volume 187 (2008) no. 1, pp. 185-186 | Article | MR 932781 | Zbl 1150.14003

[8] Cui, Y. Special loci in moduli of marked curves, Michigan Math. J., Volume 56 (2008), pp. 495-512 | Article | MR 2488722 | Zbl 1162.14016

[9] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Publications Mathématiques de l’IHES, Volume 36 (1969) no. 1, pp. 75-109 | Article | EuDML 103899 | Numdam | MR 262240 | Zbl 0181.48803

[10] Drinfelʼd, V. G. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal( ¯/), Algebra i Analiz, Volume 2 (1990) no. 4, pp. 149-181 | MR 1080203 | Zbl 0718.16034

[11] Dèbes, P.; Douai, J.-C. Algebraic covers : field of moduli versus field of definition, Annales Sci. E.N.S, Volume 30 (1997), pp. 303-338 | Numdam | MR 1443489 | Zbl 0906.12001

[12] Ekedahl, T. Boundary behaviour of Hurwitz schemes, The moduli space of curves (Texel Island, 1994) (Progr. Math.), Volume 129, Birkhäuser Boston, Boston, MA, 1995, pp. 173-198 | MR 1363057 | Zbl 0862.14018

[13] Frediani, P.; Neumann, F. Étale Homotopy Types of Moduli Stacks of Algebraic Curves with Symmetries, K-Theory (2003) no. 30, pp. 315-340 | Article | MR 2064243 | Zbl 1059.14027

[14] Fried, M. Fields of definition of function fields and Hurwitz families—groups as Galois groups, Comm. Algebra, Volume 5 (1977) no. 1, pp. 17-82 | Article | MR 453746 | Zbl 0478.12006

[15] Gonzalez-Diez, G.; Harvey, W. Fields of definition of function fields and Hurwitz families—groups as Galois groups, London Math. Soc. Lect. Note Ser., Volume 173 (1992), pp. 75-93 | Zbl 0763.32014

[16] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math., 1967 no. 32 | Numdam | Zbl 0135.39701

[17] Grothendieck, A.; Lochak, P.; Schneps, L. Esquisse d’un Programme, Geometric Galois Actions I, Volume 242, Cambridge Univ. Press, Cambridge, 1997, pp. 5-48 | MR 1483107 | Zbl 0901.14001

[18] Grothendieck, A.; Murre, J. P. The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, Lecture Notes in Mathematics, 208, Springer-Verlag, New York, 1971 | MR 316453 | Zbl 0216.33001

[19] Ihara, Y.; Schneps, L.; Lochak, P. On the embedding of Gal( ¯/) into GT ^, The Grothendieck Theory of Dessins d’Enfants, Volume 200, Cambridge Univ. Press, Cambridge, 1994, pp. 289-321 | MR 1305402

[20] Kerckhoff, S. P. The Nielsen realization problem, Ann. of Math. (2), Volume 117 (1983) no. 2, pp. 235-265 | Article | MR 690845 | Zbl 0528.57008

[21] Knudsen, F. F. The projectivity of the moduli space of stable curves. II. The stacks M g,n , Math. Scand., Volume 52 (1983) no. 2, pp. 161-199 | MR 702953 | Zbl 0544.14020

[22] Lochak, P. Results and conjectures in profinite Teichmüller theory, Galois-Teichmüller theory and arithmetic geometry (Adv. Stud. Pure Math.), Volume 63, Math. Soc. Japan, Tokyo, 2012, pp. 263-335 | MR 3051247

[23] Lochak, P.; Schneps, L.; symp, proc. Open problems in Grothendieck-Teichmüller theory, Amer. Math. Soc., 2006, pp. 165-186 | MR 2264540 | Zbl 1222.14046

[24] Maugeais, S. Quelques déformations sur les déformations équivariantes des courbes stables, Manuscripta Math. (2006) no. 120, pp. 53-82 | Article | MR 2223481 | Zbl 1101.14038

[25] Mumford, D. Abelian quotients of the Teichmüller modular group, Journal d’Analyse Mathématique, Volume 18 (1967) no. 1, pp. 227-244 | Article | MR 219543 | Zbl 0173.22903

[26] Nakamura, H. Galois rigidity of pure sphere braid groups and profinite calculus, Journal Mathematical Sciences University Tokyo, Volume 1 (1994), pp. 71-136 | MR 1298541 | Zbl 0901.14012

[27] Nakamura, H. Galois representations in the profinite Teichmüller modular groups, Geometric Galois actions, 1 (London Math. Soc. Lecture Note Series), Cambridge Univ. Press, Cambridge, 1997, pp. 159-174 | MR 1483116 | Zbl 0911.14014

[28] Nakamura, H. Limits of Galois representations in fundamental groups along maximal degeneration of marked curves. I, Amer. J. Math., Volume 121 (1999) no. 2, pp. 315-358 | Article | MR 1680325 | Zbl 1006.12001

[29] Nakamura, H.; Schneps, L. On a subgroup of the Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups, Inventiones mathematica, Volume 141 (2000) no. 1, pp. 503-560 | Article | MR 1779619 | Zbl 1077.14030

[30] Noohi, B. Fundamental groups of algebraic stacks, Journal of the Institute of Mathematics of Jussieu, Volume 3 (2004) no. 01, pp. 69-103 | Article | MR 2036598 | Zbl 1052.14001

[31] Oda, T. Etale homotopy type of the moduli spaces of algebraic curves, Geometric Galois actions, 1 (London Math. Soc. Lecture Note Ser.), Volume 242, Cambridge Univ. Press, Cambridge, 1997, pp. 85-95 | MR 1483111 | Zbl 0902.14019

[32] Romagny, M. Composantes connexes et irréductibles en familles, Manuscripta Math., Volume 136 (2011) no. 1-2, pp. 1-32 | Article | MR 2820394 | Zbl 1266.14010

[33] Schneps, L. Special loci in moduli spaces of curves, Galois groups and fundamental groups, Volume 41, Cambridge Univ. Press, Cambridge, 2003 | MR 2012218 | Zbl 1071.14028

[34] Serre, J.-P. Two letters on non-abelian cohomology, Geometric galois actions : around Grothendieck’s esquisse d’un programme, Cambridge Univ. Press, 1997 | MR 1483117 | Zbl 0886.20035

[35] Symonds, P. On cohomology isomorphisms of groups, J. Algebra, Volume 313 (2007) no. 2, pp. 802-810 | Article | MR 2329570 | Zbl 1131.20038

[36] Tufféry, S. Déformations de courbes avec action de groupe, Forum Math., Volume 5 (1993) no. 3, pp. 243-259 | MR 1216034 | Zbl 0809.14005

[37] Zoonekynd, V. La tour de Teichmüller-Grothendieck (2001) (Ph. D. Thesis)

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