Location of Repository

Wilson's map operations on regulat dessins and cyclotomic fields of definition

By Gareth Jones, M. Streit and J. Wolfart

Abstract

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

Suggested articles

Preview

Citations

  1. (2004). ABC for polynomials, dessins d’enfants, and uniformization—a survey’, Elementare 965 und Analytische Zahlentheorie (Tagungsband),
  2. (1971). Automorphisms of imbedded graphs’, doi
  3. (1979). Bely˘ ı, ‘On Galois extensions of a maximal cyclotomic field’, doi
  4. (1996). Belyi functions, hypermaps and Galois groups’, doi
  5. (2000). Characters and Galois invariants of regular dessins’, R e v .M a t .C o m p l u t .13 doi
  6. (2001). Cyclic projective planes and Wada dessins’,
  7. (1990). Edge-symmetric orientable imbeddings of complete graphs’, doi
  8. (1967). Endliche Gruppen I (Springer, doi
  9. (1989). Equilateral triangulations of Riemann surfaces and curves over algebraic number fields’,
  10. (1997). Esquisse d’un Programme’, Geometric Galois actions 1. Around Grothendieck’s Esquisse d’un Programme doi
  11. (2000). Field of definition and Galois orbits for the Macbeath–Hurwitz curves’, doi
  12. (1978). Finitely maximal Fuchsian groups’, doi
  13. (1994). Fuchsian triangle groups and Grothendieck dessins: variations on a theme of Belyi’, doi
  14. (2007). Galois action on families of generalised Fermat curves’, doi
  15. (1997). Galois groups, monodromy groups and cartographic groups’, Geometric Galois actions 2. The inverse Galois problem, moduli spaces and mapping class groups doi
  16. (2009). Generalised Fermat hypermaps and Galois orbits’, doi
  17. (1980). Generators and relations for discrete groups (Springer, doi
  18. (1967). Generators of the linear fractional groups’, Number theory,( e d .W .J .L e v e q u ea n d E. doi
  19. (1985). Hurwitz families and arithmetic Galois groups’, doi
  20. (1975). Hypermaps versus bipartite maps’, doi
  21. (1999). On fields of moduli of curves’, doi
  22. (1984). Operations on maps, and outer automorphisms’, doi
  23. (1979). Operators over regular maps’, doi
  24. (2000). Regular cyclic coverings of the Platonic maps’, doi
  25. (2007). Regular embeddings of Kn,n where n is an odd prime power’, doi
  26. (1985). Regular orientable imbeddings of complete graphs’, doi
  27. (1997). The ‘obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms’, doi
  28. (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.