JournalsrmiVol. 29, No. 1pp. 163–181

Galois actions on regular dessins of small genera

  • Marston D. E. Conder

    University of Auckland, New Zealand
  • Gareth A. Jones

    University of Southampton, UK
  • Manfred Streit

    Oberursel, Germany
  • Jürgen Wolfart

    Universität Frankfurt, Germany
Galois actions on regular dessins of small genera cover
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Abstract

Dessins d'enfants can be regarded as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve defined over a number field). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge-transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the fields of definition for all curves with regular dessins of genus 2 to 18.

Cite this article

Marston D. E. Conder, Gareth A. Jones, Manfred Streit, Jürgen Wolfart, Galois actions on regular dessins of small genera. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 163–181

DOI 10.4171/RMI/717