Galois actions on regular dessins of small genera
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.
scientific article; zbMATH DE number 6143291