Comparison of two notions of rationality for a dessin d'enfant
type: Item
instance of: scholarly article
title: Comparison of two notions of rationality for a dessin d'enfant
zbMATH Open document ID: 1002.11056
author: Layla Pharamond dit d'Costa
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2002-07-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/225299
full work available at URL: jourelec
review text: Let \(f:{\mathcal C}\to \mathbb{P}^1\) be a ramified covering, where \({\mathcal C}\) is an algebraic curve, defined over an algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\). Let \(\sigma\) be an element of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\), denote by \({}^\sigma f: {}^\sigma{\mathcal C}\to \mathbb{P}^1\) the covering obtained by base change. There are two ways of saying that the covering is ``stable'' under \(\sigma\): 1) there exists \(u_\sigma:{\mathcal C}\simeq {}^\sigma{\mathcal C}\), defined over \(\overline{\mathbb{Q}}\), such that \(f= {}^\sigma fu_\sigma\), 2) there exists \(u_\sigma\) as above and an automorphism \(v_\sigma\) of \(\mathbb{P}^1\), defined over \(\overline{\mathbb{Q}}\), such that \(v_\sigma f= {}^\sigma fu_\sigma\). In both cases, the set of such \(\sigma\) is an open subgroup of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\), then one can define ``fields of definition'' for the covering \(f\). The author compares these two natural notions of rationality, especially when they differ, when the two fields of definition are not the same. Explicit examples are given.
reviewed by: Marc Reversat
Mathematics Subject Classification ID: 11G99
Mathematics Subject Classification ID: 14G32
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 1770784
zbMATH Keywords: Galois group
zbMATH Keywords: fields of definition
zbMATH Keywords: ramified covering
zbMATH Keywords: rationality
MaRDI profile type: Publication
OpenAlex ID: W2316785144
cites work: Algebraic covers: Field of moduli versus field of definition
cites work: Q4364755
cites work: Q4263461
cites work: Q4003835
cites work: The Field of Definition of a Variety
DOI: 10.5802/JTNB.336
description: scientific article; zbMATH DE number 1770784