Location of Repository

Dessins d’enfants can be seen as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, even an algebraic structure as a projective algebraic curve defined over a number field. Combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve, such as the field of moduli or the field of definition. However, apart from a few series of examples, very few general results concerning such correspondences are known. As a step in this direction, we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields, based on Wilson's operations on maps: these leave invariant the graph but change the cyclic ordering of edges around the vertices, therefore they change the embeddings, and hence the dessins, and hence the conformal and algebraic structure of the underlying curves. Under suitable assumptions, satisfied by many series of regular dessins, these changes coincide with the effect of Galois conjugation. This coincidence allows one to draw conclusions about Galois orbits and fields of definition of dessins. The possibilities afforded by these techniques, and their limitations, are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs. <br/

Year: 2009

OAI identifier:
oai:eprints.soton.ac.uk:156469

Provided by:
e-Prints Soton

Downloaded from
http://dx.doi.org/10.1112/plms/pdp033

- (2004). ABC for polynomials, dessins d’enfants, and uniformization—a survey’, Elementare 965 und Analytische Zahlentheorie (Tagungsband),
- (1971). Automorphisms of imbedded graphs’,
- (1979). Bely˘ ı, ‘On Galois extensions of a maximal cyclotomic ﬁeld’,
- (1996). Belyi functions, hypermaps and Galois groups’,
- (2000). Characters and Galois invariants of regular dessins’, R e v .M a t .C o m p l u t .13
- (2001). Cyclic projective planes and Wada dessins’,
- (1990). Edge-symmetric orientable imbeddings of complete graphs’,
- (1967). Endliche Gruppen I (Springer,
- (1989). Equilateral triangulations of Riemann surfaces and curves over algebraic number ﬁelds’,
- (1997). Esquisse d’un Programme’, Geometric Galois actions 1. Around Grothendieck’s Esquisse d’un Programme
- (2000). Field of deﬁnition and Galois orbits for the Macbeath–Hurwitz curves’,
- (1978). Finitely maximal Fuchsian groups’,
- (1994). Fuchsian triangle groups and Grothendieck dessins: variations on a theme of Belyi’,
- (2007). Galois action on families of generalised Fermat curves’,
- (1997). Galois groups, monodromy groups and cartographic groups’, Geometric Galois actions 2. The inverse Galois problem, moduli spaces and mapping class groups
- (2009). Generalised Fermat hypermaps and Galois orbits’,
- (1980). Generators and relations for discrete groups (Springer,
- (1967). Generators of the linear fractional groups’, Number theory,( e d .W .J .L e v e q u ea n d E.
- (1985). Hurwitz families and arithmetic Galois groups’,
- (1975). Hypermaps versus bipartite maps’,
- (1999). On ﬁelds of moduli of curves’,
- (1984). Operations on maps, and outer automorphisms’,
- (1979). Operators over regular maps’,
- (2000). Regular cyclic coverings of the Platonic maps’,
- (2007). Regular embeddings of Kn,n where n is an odd prime power’,
- (1985). Regular orientable imbeddings of complete graphs’,
- (1997). The ‘obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms’,
- (1975). Un code pour les graphes planaires et ses applications’ Ast´ erisque 27 (Soci´ et´ eM a t h ´ ematique de France,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.