Galois covers and the Hilbert-Grunwald property
type: Item
instance of: scholarly article
title: Galois covers and the Hilbert-Grunwald property
zbMATH Open document ID: 1255.14022
author: Pierre Dèbes
author: Nour Ghazi
publication date: 2012-10-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: Let \(K\) be a field, \(S\) a finite set of independent, non-trivial discrete valuations of \(K\) and \(G\) a finite group. Given a family \((E_v/K_v)_{v \in S}\) of Galois extensions of the \(v\)-completions \(K_v\) of \(K\), a solution to the Grunwald problem is a Galois extension \(L/K\) such that for all \(v \in S\) its local extensions are isomorphic to \(E_v/K_v\). The main result of this paper (Theorem 1.2) considers the case that \(K\) is an algebraic number field. It shows that a Galois cover \(f: X \to \mathbb P^1\) with group \(G\) and some good-reduction condition imply that each unramified Grunwald problem has solutions which are specializations of \(f\) at points in \(\mathbb A^1(K)\) outside the branch divisor. The authors deduce this result from a more general one (Theorem 3.2), which considers an arbitrary field \(K\) and a \(G\)-cover over a smooth projective \(K\)-variety with some further integrality conditions. For the proof, the authors employ a ``Twisting Lemma'' (2.1) to obtain the result in the local situation (Proposition 2.2), and then globalize (Chapter 3). The results have connections to Hilbert's Irreducibility Theorem and the Regular Inverse Galois problem.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 12E25
zbMATH DE Number: 6093170
zbMATH Keywords: algebraic covers
zbMATH Keywords: Hurwitz spaces
zbMATH Keywords: Grunwald's problem
zbMATH Keywords: inverse Galois theory
MaRDI profile type: Publication
OpenAlex ID: W2507888222
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DOI: 10.5802/AIF.2714
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6093170