Flon, Stéphane Ramification in the field of moduli. (Ramification dans les corps des modules.) (French) Zbl 1066.14016 Ann. Inst. Fourier 54, No. 2, 253-263 (2004). Let \(f\) be an algebraic cover of the projective line, defined over \(\overline{\mathbb Q}\), with monodromy group \(G\) and let \(K\) be the compositum of residue fields of branch points of \(f\) and \(M\) the corresponding field of moduli. A result of S. Beckmann [J. Algebra 125, 236–255 (1989; Zbl 0698.14024)] gives information on the ramifications of bad places of \(f\) that divide the order of \(G\). In this paper, the author studies the ramification of bad places of \(f\) that do not divide the order of \(G\), but for which the branch points of \(f\) meet when reduced. This is done by studying geometric and arithmetic aspects of completions of Hurwitz spaces. Reviewer: Christodor-Paul Ionescu (Bucureşti) Cited in 3 Documents MSC: 14E22 Ramification problems in algebraic geometry 14H30 Coverings of curves, fundamental group 12F12 Inverse Galois theory Keywords:completion of Hurwitz spaces; inverse Galois problem Citations:Zbl 0698.14024 PDFBibTeX XMLCite \textit{S. Flon}, Ann. Inst. Fourier 54, No. 2, 253--263 (2004; Zbl 1066.14016) Full Text: DOI Numdam EuDML References: [1] Ramified primes in the field of moduli of branched coverings of curves, J. Algebra, 125, 236-255 (1989) · Zbl 0698.14024 [2] Braids, links, and mapping class groups, vol. 82 (1974) · Zbl 0305.57013 [3] Dessins from a geometric point of view, The Grothendieck theory of dessins d’enfants, 79-113 (1995) · Zbl 0835.14010 [4] Hurwitz families and arithmetic Galois groups, Duke Math. J., 52, 821-839 (1985) · Zbl 0601.14023 [5] Calcul et rationalité de fonctions de Belyi en genre \(0\), Ann. Inst. Fourier, 44, 1, 1-38 (1994) · Zbl 0791.11059 [6] Groupes de Galois sur \(K(T)\), Séminaire de théorie des nombres de Bordeaux, 2, 229-243 (1990) · Zbl 0729.12010 [7] Covers of \(\mathbb{P}^1\) over the \(p\)-adics, Contemp. Math., 186, 217-238 (1995) · Zbl 0856.12004 [8] Algebraic covers: Field of moduli versus field of definition, Ann. Sci. École Normale Sup., 30, 303-338 (1997) · Zbl 0906.12001 [9] Nonrigid constructions in Galois theory, Pacific J. Math., 56, 81-122 (1990) · Zbl 0788.12001 [10] On reduction of covers of arithmetic surfaces, Contemp. Math., 245, 117-132 (1999) · Zbl 0978.14012 [11] Sur les espaces de Hurwitz, Séminaires et congrès, 5, 69-99 (2001) · Zbl 1078.14521 [12] Bonnes places et corps des modules, Séminaires et congrès, 5, 101-117 (2001) · Zbl 1078.14510 [13] Mauvaises places ramifiées dans le corps des modules d’un revêtement (2002) [14] Restriction de revêtements (2002) [15] Fields of definition of function fields and Hurwitz families. Groups as Galois groups, Comm. in Alg., 1, 17-82 (1977) · Zbl 0478.12006 [16] Hurwitz schemes and the irreducibility of the moduli of algebraic curves, Ann. Math., 90, 542-575 (1969) · Zbl 0194.21901 [17] The inverse Galois problem and rational points on moduli spaces, Math. Ann., 290, 771-800 (1991) · Zbl 0763.12004 [18] Presentation and central extensions of mapping class groups, Trans. Amer. Math. Soc., 348, 8, 3097-3132 (1996) · Zbl 0861.57023 [19] Stable \(n\)-pointed trees of projective lines, Proc. Koningklijke Nederlandse Akademie van Wetenschappen, 91, 131-163 (1988) · Zbl 0698.14019 [20] The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, vol. 208 (1971) · Zbl 0216.33001 [21] Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., 39, 1-61 (1891) · JFM 23.0429.01 [22] A presentation of the mapping class group, Math. Res. Lett., 4, 735-739 (1997) · Zbl 0891.20028 [23] Primary units and a proof of Catalan’s conjecture (2003) [24] Inverse Galois theory (1999) · Zbl 0940.12001 [25] Construction of Hurwitz spaces (1998) · Zbl 0925.14002 [26] Deformation of tame admissible covers of curves (1998) · Zbl 0995.14008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.