Galois Covers and the Hilbert-Grunwald Property

Dèbes, Pierre; Ghazi, Nour

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 989-1013.

Notre résultat principal mêle plusieurs thèmes  : il contient une conclusion de type Grunwald-Wang, une version du théorème d’irréductibilité de Hilbert et une forme $p$-adique à la Harbater, mais avec bonne réduction, du problème inverse de Galois sous sa forme régulière (RIGP). Nous en déduisons un énoncé qui pose de nouvelles questions sur le RIGP sur $\mathbb{Q}$. La stratégie générale est d’étudier et d’exploiter la bonne réduction de certains modèles tordus de revêtements et des espaces de modules associés.

Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a $p$-adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over $\mathbb{Q}$. The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.

Classification : 14H30, 11R32, 12F12, 12E25, 14Gxx, 14Dxx, 14H10
Keywords: Inverse Galois theory, Grunwald’s problem, Hilbert’s irreducibility theorem, algebraic covers, local and global fields, Hurwitz spaces
Mot clés : théorie inverse de Galois, problème de Grunwald, théorème d’irréductibilité de Hilbert, revêtements algébriques, corsp locaux, corps globaux, espaces de Hurwitz

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