Ramification dans le corps des modules  [ Ramification in the field of moduli ]
Annales de l'Institut Fourier, Volume 54 (2004) no. 2, p. 253-293
Let f be an algebraic cover of the projective line, defined over ¯, with monodromy group G. Let K be the compositum of residue fields of branch points of f, and let M be the corresponding field of moduli. Starting from the link between field of moduli and Hurwitz spaces, we study both geometric and arithmetic aspects of the (completions of) Hurwitz spaces and configuration spaces, in order to estimate ramification of bad places of f that do not divide |G|, but for which the branch points of f meet when reduced. We then discuss the possibility to bound ramification in the field of moduli to places that do not divide |G|, just by choosing an appropriate branch locus.
Soit f un revêtement de la droite projective défini sur ¯, de groupe de monodromie G. Soit K le compositum des corps de rationalité des points de branchement f, et M le corps des modules correspondants. Partant du lien entre corps des modules et espaces de Hurwitz, on étudie la géométrie et l’arithmétique de ces espaces et des espaces de configuration de points complétés pour évaluer la ramification dans M/K des mauvaises places de f qui ne divisent pas l’ordre de G, mais où les points de branchements de f se rencontrent par réduction. On discute enfin la possibilité de limiter, par choix du lieu de branchement, la ramification dans les corps des modules aux seules places divisant |G|.
DOI : https://doi.org/10.5802/aif.2018
Classification:  14D22,  14E22,  14H30
Keywords: field of moduli, completions of Hurwitz spaces, (G)-covers, inverse Galois problem , ABC conjecture
@article{AIF_2004__54_2_253_0,
     author = {Flon, St\'ephane},
     title = {Ramification dans le corps des modules},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {2},
     year = {2004},
     pages = {253-293},
     doi = {10.5802/aif.2018},
     zbl = {1066.14016},
     mrnumber = {2073835},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2004__54_2_253_0}
}
Flon, Stéphane. Ramification dans le corps des modules. Annales de l'Institut Fourier, Volume 54 (2004) no. 2, pp. 253-293. doi : 10.5802/aif.2018. http://www.numdam.org/item/AIF_2004__54_2_253_0/

[Be] S. Beckmann Ramified primes in the field of moduli of branched coverings of curves, J. Algebra, Tome 125 (1989), pp. 236-255 | MR 1012673 | Zbl 0698.14024

[Bi] J. Birman Braids, links, and mapping class groups, Princeton Univ. Press, Annals of Math. Studies, Tome vol. 82 (1974) | MR 375281 | Zbl 0305.57013

[CG] J.-M. Couveignes; L. Granboulan; Leila Schneps Ed. Dessins from a geometric point of view, The Grothendieck theory of dessins d'enfants, Cambridge University Press (1995), pp. 79-113 | MR 1305394 | Zbl 0835.14010

[CH] K. Coombes; D. Harbater Hurwitz families and arithmetic Galois groups, Duke Math. J., Tome 52 (1985), pp. 821-839 | MR 816387 | Zbl 0601.14023

[Co] J.-M. Couveignes Calcul et rationalité de fonctions de Belyi en genre 0, Ann. Inst. Fourier, Tome 44 (1994) no. 1, pp. 1-38 | Numdam | MR 1262878 | Zbl 0791.11059

[D1] P. Dèbes Groupes de Galois sur K(T), Séminaire de théorie des nombres de Bordeaux, Tome 2 (1990), pp. 229-243 | Numdam | MR 1081725 | Zbl 0729.12010

[D2] P. Dèbes Covers of 1 over the p-adics, Contemp. Math., Tome 186 (1995), pp. 217-238 | MR 1352273 | Zbl 0856.12004

[DD] P. Dèbes; J.-C. Douai Algebraic covers: Field of moduli versus field of definition, Ann. Sci. École Normale Sup., Tome 30 (1997), pp. 303-338 | Numdam | MR 1443489 | Zbl 0906.12001

[DF] P. Dèbes; M. Fried Nonrigid constructions in Galois theory, Pacific J. Math., Tome 56 (1990), pp. 81-122 | MR 1256178 | Zbl 0788.12001

[E1] M. Emsalem On reduction of covers of arithmetic surfaces, Contemp. Math., Tome 245 (1999), pp. 117-132 | MR 1732232 | Zbl 0978.14012

[E2] M. Emsalem Sur les espaces de Hurwitz, Séminaires et congrès, Tome 5 (2001), pp. 69-99 | Zbl 01974559

[F1] S. Flon Bonnes places et corps des modules, Séminaires et congrès, Tome 5 (2001), pp. 101-117 | MR 1924918 | Zbl 01974560

[F2] S. Flon Mauvaises places ramifiées dans le corps des modules d'un revêtement (2002) (Université des Sciences et Technologies de Lille)

[F3] S. Flon Restriction de revêtements (2002) (Université des Sciences et Technologies de Lille)

[Fr] M. Fried Fields of definition of function fields and Hurwitz families. Groups as Galois groups, Comm. in Alg., Tome 1 (1977), pp. 17-82 | MR 453746 | Zbl 0478.12006

[Fu] W. Fulton Hurwitz schemes and the irreducibility of the moduli of algebraic curves, Ann. Math., Tome 90 (1969), pp. 542-575 | MR 260752 | Zbl 0194.21901

[FV] M. Fried; H. Völklein The inverse Galois problem and rational points on moduli spaces, Math. Ann., Tome 290 (1991), pp. 771-800 | MR 1119950 | Zbl 0763.12004

[Ge] S. Gervais Presentation and central extensions of mapping class groups, Trans. Amer. Math. Soc., Tome 348 (1996) no. 8, pp. 3097-3132 | MR 1327256 | Zbl 0861.57023

[GHP] L. Gerritzen; F. Herrlich; M. Van Der Put Stable n-pointed trees of projective lines, Proc. Koningklijke Nederlandse Akademie van Wetenschappen, Tome 91 (1988), pp. 131-163 | MR 952512 | Zbl 0698.14019

[GM] A. Grothendieck; J.P. Murre The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Springer-Verlag Tome vol. 208 (1971) | MR 316453 | Zbl 0216.33001

[Hu] A. Hurwitz Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Tome 39 (1891), pp. 1-61 | JFM 23.0429.01 | MR 1510692

[Lu] F. Luo A presentation of the mapping class group, Math. Res. Lett., Tome 4 (1997), pp. 735-739 | MR 1484704 | Zbl 0891.20028

[Mi] P. Mihăilescu Primary units and a proof of Catalan's conjecture (2003) (soumis au Jounal de Crelle)

[MM] G. Malle; B.H. Matzat Inverse Galois theory, Springer-Verlag (1999) | MR 1711577 | Zbl 0940.12001

[W1] S. Wewers Construction of Hurwitz spaces (1998) (Thèse, Université d'Essen) | Zbl 0925.14002

[W2] S. Wewers Deformation of tame admissible covers of curves, Institut für Experimentelle Mathematik Essen (1998) | MR 1708609 | Zbl 0995.14008