We continue the examination of the stable reduction and fields of moduli of -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic , where has a cyclic -Sylow subgroup of order . Suppose further that the normalizer of acts on via an involution. Under mild assumptions, if is a three-point -Galois cover defined over , then the th higher ramification groups above for the upper numbering of the (Galois closure of the) extension vanish, where is the field of moduli of .
Nous poursuivons l’étude de la réduction stable et des corps de modules des -revêtements galoisiens de la droite projective sur un corps discrètement valué de caractéristique mixte , dans le cas où a un -sous-groupe de Sylow cyclique d’ordre . Supposons de plus que le normalisateur de agit sur lui-même via une involution. Sous des hypothèses assez légères, nous montrons que si est un -revêtement galoisien ramifié au-dessus de points, défini sur , alors les -ièmes groupes de ramification supérieure au-dessus de , en numérotation supérieure, de (la clôture galoisienne de) l’extension sont triviaux, où est le corps des modules de .
Mots-clés : field of moduli, stable reduction, Galois cover
Andrew Obus 1
@article{JTNB_2013__25_3_579_0, author = {Andrew Obus}, title = {Fields of moduli of three-point $G$-covers with cyclic $p${-Sylow,} {II}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {579--633}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {25}, number = {3}, year = {2013}, doi = {10.5802/jtnb.850}, mrnumber = {3179678}, zbl = {06291369}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.850/} }
TY - JOUR AU - Andrew Obus TI - Fields of moduli of three-point $G$-covers with cyclic $p$-Sylow, II JO - Journal de théorie des nombres de Bordeaux PY - 2013 SP - 579 EP - 633 VL - 25 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.850/ DO - 10.5802/jtnb.850 LA - en ID - JTNB_2013__25_3_579_0 ER -
%0 Journal Article %A Andrew Obus %T Fields of moduli of three-point $G$-covers with cyclic $p$-Sylow, II %J Journal de théorie des nombres de Bordeaux %D 2013 %P 579-633 %V 25 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.850/ %R 10.5802/jtnb.850 %G en %F JTNB_2013__25_3_579_0
Andrew Obus. Fields of moduli of three-point $G$-covers with cyclic $p$-Sylow, II. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 579-633. doi : 10.5802/jtnb.850. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.850/
[1] S. Beckmann, Ramified primes in the field of moduli of branched coverings of curves. J. Algebra 125 (1989), 236–255. | MR | Zbl
[2] I. I. Bouw and S. Wewers, Reduction of covers and Hurwitz spaces. J. Reine Angew. Math. 574 (2004), 1–49. | MR | Zbl
[3] K. Coombes and D. Harbater, Hurwitz families and arithmetic Galois groups. Duke Math. J. 52 (1985), 821–839. | MR | Zbl
[4] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. | Numdam | MR | Zbl
[5] P. Deligne and M. Rapoport, Les schémas de modules de courbes élliptiques. Modular functions of one variable II, LNM 349, Springer-Verlag (1972), 143–316. | MR | Zbl
[6] W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. of Math. 90 (1969), no. 2, 542–575. | MR | Zbl
[7] Y. Henrio, Disques et couronnes ultramétriques. Courbes semi-stables et groupe fondamental en géométrie algébrique, Progr. Math., 187, Birkhäuser Verlag, Basel (1998), 21–32. | MR | Zbl
[8] Y. Henrio, Arbres de Hurwitz et automorphismes d’ordre des disques et des couronnes -adiques formels. arXiv:math/0011098
[9] B. Huppert, Endliche gruppen. Springer-Verlag, Berlin, 1987. | Zbl
[10] N. Katz, Local-to-global extensions of fundamental groups. Ann. Inst. Fourier, Grenoble 36 (1986), 69–106. | Numdam | MR | Zbl
[11] G. Malle and B. H. Matzat, Inverse Galois theory. Springer-Verlag, Berlin, 1999. | MR | Zbl
[12] A. Obus, Fields of moduli of three-point -covers with cyclic -Sylow, I. Algebra Number Theory 6 (2012), no. 5, 833–883. | MR | Zbl
[13] A. Obus, Toward Abhyankar’s inertia conjecture for . Groupes de Galois géométriques et différentiels, Séminaires et Congrès, 27, Société Mathématique de France (2013), 191–202.
[14] A. Obus, Conductors of extensions of local fields, especially in characteristic . To appear in Proc. Amer. Math. Soc.
[15] A. Obus, Vanishing cycles and wild monodromy. Int. Math. Res. Notices (2012), 299–338. | MR
[16] A. Obus and S. Wewers, Cyclic extensions and the local lifting problem. To appear in Ann. of Math.
[17] R. Pries, Wildly ramified covers with large genus. J. Number Theory 119 (2006), 194–209. | MR | Zbl
[18] M. Raynaud, Revêtements de la droite affine en caractéristique et conjecture d’Abhyankar. Invent. Math. 116 (1994), 425–462. | MR | Zbl
[19] M. Raynaud, Specialization des revêtements en caractéristique . Ann. Sci. École Norm. Sup. 32 (1999), 87–126. | Numdam | MR | Zbl
[20] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 1–42. | Numdam | MR | Zbl
[21] J.-P. Serre, Local fields. Springer-Verlag, New York, 1979. | MR | Zbl
[22] S. Wewers, Reduction and lifting of special metacyclic covers. Ann. Sci. École Norm. Sup. (4) 36 (2003), 113–138. | Numdam | MR | Zbl
[23] S. Wewers, Three point covers with bad reduction. J. Amer. Math. Soc. 16 (2003), 991–1032. | MR | Zbl
[24] S. Wewers, Formal deformation of curves with group scheme action. Ann. Inst. Fourier 55 (2005), 1105-1165. | Numdam | MR | Zbl
[25] H. J. Zassenhaus, The theory of groups, 2nd ed.. Chelsea Publishing Company, New York, 1956. | MR | Zbl
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