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Mauvaises places ramifiées dans le corps des modules d'un revêtement

Abstract : This work is based on the link between the field of moduli of a cover and Hurwitz spaces. Given a cover, arithmetic aspects of these spaces provide results about ramification in the field of moduli over the rationality field of the branch points. For instance, Beckmann's theorem, which limits search of ramification in this extension to some places, namely the bad places, is proved in a natural way in this context. A finer analysis of Hurwitz spaces gives information about the bad places that do not divide the order of the monodromy group of the cover (but for which the branch locus becomes singular, though) : the point is to consider the cover of the completion of the Hurwitz space over the completion of the configuration space. Given any such place, the branch locus extends to an arithmetic section on this latter space, and restriction of the Hurwitz cover provides data on the ramification of the chosen place in the field of moduli. We study this restriction problem in a more general context, in which we consider the case of the restriction to a section of a tamely ramified cover over a divisor with normal crossings, using Abhyankar's theorem. In order to give an explicit version of this ramification result on the field of moduli, we introduce integers that depend on the coalescing orders between branch points, and a choice of inertia generators around the boundary components of the configuration space that meet the given arithmetic section. For this purpose, we define a certain type of Dehn twists, the so called "twists sarments", and we describe their action on the set of Nielsen classes. Last part of this work contains various results of descent of the field of definition of a cover, by studying gerbs over Hurwitz spaces (and not merely these coarse moduli spaces).
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https://tel.archives-ouvertes.fr/tel-00002259
Contributor : Stéphane Flon <>
Submitted on : Friday, January 10, 2003 - 5:12:45 PM
Last modification on : Saturday, April 20, 2019 - 1:23:48 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 7:20:17 PM

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  • HAL Id : tel-00002259, version 1

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Stéphane Flon. Mauvaises places ramifiées dans le corps des modules d'un revêtement. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2002. Français. ⟨tel-00002259⟩

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