Revista Matemática Iberoamericana

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Volume 29, Issue 1, 2013, pp. 163–181
DOI: 10.4171/RMI/717

Published online: 2013-01-14

Galois actions on regular dessins of small genera

Marston D. E. Conder[1], Gareth A. Jones[2], Manfred Streit[3] and Jürgen Wolfart[4]

(1) University of Auckland, New Zealand
(2) University of Southampton, UK
(3) Oberursel, Germany
(4) Universität Frankfurt, Germany

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.

Keywords: Regular maps and hypermaps, dessins d'enfants, Galois actions, fields of definition

Conder Marston, Jones Gareth, Streit Manfred, Wolfart Jürgen: Galois actions on regular dessins of small genera. Rev. Mat. Iberoamericana 29 (2013), 163-181. doi: 10.4171/RMI/717