Effective descent of the field of definition of a Galois cover
type: Item
instance of: scholarly article
title: Effective descent of the field of definition of a Galois cover
zbMATH Open document ID: 0945.12003
author: Bounab Sadi
published in: Journal of Number Theory
publication date: 2000-10-08T00:00:00Z
review text: Let \(k\) be a field of zero characteristic and let \(f:X\to \mathbb{P} {1\over k}\) be a cover defined over an algebraic closure \(\overline k\) of \(k\). Write \(f^\tau: X^\tau\to \mathbb{P}{1 \over k}\) for the cover obtained from \(f\) by the action of an element \(\tau\) of the Galois group \(G=\text{Gal}(\overline k/k)\). The field of moduli of \(f\) is \(k_0=\overline k^H\), where \(H\) is the subgroup of elements \(\tau\) of \(G\) such that \(f\simeq f^\tau\). This field is contained in any field of definition of the cover \(f\). When \(f\) is a Galois cover, \textit{K. R. Coombes} and \textit{D. Harbater} have proved that \(f\) is defined over \(k_0\) [Duke Math. J. 52, 821-839 (1985; Zbl 0601.14023)]. A new proof of this result is given by the author, it has the advantage to give an explicit model (with bounds for the degrees and the heights of its equations) of the Galois cover \(f\) over its fields of moduli \(k_0\).
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 1309268
zbMATH Keywords: Galois cover
zbMATH Keywords: cover
zbMATH Keywords: field of moduli
zbMATH Keywords: field of definition
reviewed by: Marc Reversat
MaRDI profile type: Publication
full work available at URL: jnth.1998.2368
OpenAlex ID: W2084216348
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DOI: 10.1006/JNTH.1998.2368
description: scientific article; zbMATH DE number 1309268