Galois actions on regular dessins of small genera
type: Item
instance of: scholarly article
title: Galois actions on regular dessins of small genera
zbMATH Open document ID: 1262.14034
DOI: 10.4171/RMI/717
published in: Revista Matemática Iberoamericana
publication date: 2013-03-11T00:00:00Z
review text: The dessins d'enfants have several definitions in algebraic or topological terms. One of them is to consider the dessins as bipartite graphs embedded in compact orientable surfaces. Then a dessin determines a complex structure on the surface, as well as an algebraic structure as a projective algebraic curve defined over a number field. The paper under review deals with the problem of determining the properties of the curve from the combinatorial properties of the dessin. The authors restrict themselves to regular dessins, that is to say, those having an edge-transitive, colour-preserving, orientation-preserving automorphism group. This automorphism group is studied by means of the uniformization of Riemann surfaces by Fuchsian groups. Two general results arise, namely Theorem 1 and Proposition 3 in the paper, according to which, if a regular dessin is determined up to isomorphism by genus, signature (of the Fuchsian group) and automorphism group, or else the automorphism group is Abelian, then the dessin can be defined over \(\mathbb{Q}\). Then the authors go to the classification in [\textit{M. D. E. Conder}, J. Comb. Theory, Ser. B 99, No. 2, 455--459 (2009; Zbl 1205.05106)] and consider all regular dessins on curves of genera 2 to 18, excluding those satisfying the conditions in Theorem 1 or Proposition 3. They obtain a table, which gives for each dessin the genus, the signature, the order of the automorphism group, the number of non-isomorphic curves for the given dessin, and the minimal fields of definition of the curve and of the dessin. This information is justified with a detailed case-by-case study. The paper ends stating and commenting the problems that appear in higher genera, as well as the possible ways in order to study them. All in all, it is a paper with a deep information and important new ideas to develop.
Mathematics Subject Classification ID: 14H57
Mathematics Subject Classification ID: 11G32
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 6143291
zbMATH Keywords: regular maps and hypermaps
zbMATH Keywords: dessins d'enfants
zbMATH Keywords: Galois actions
zbMATH Keywords: fields of definition
author: Marston D. E. Conder
author: Gareth Jones
author: Manfred Streit
author: Jürgen Wolfart
reviewed by: José Javier Etayo
describes a project that uses: Magma
describes a project that uses: GAP
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6143291