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

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

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.

510-532

Jones, Gareth

fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5

Streit, M.

d0b49a1c-0574-4062-bd68-49a0b4442cad

Wolfart, J.

7824cced-2f37-453b-b8b5-29647679aab3

25 August 2009

Jones, Gareth

fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5

Streit, M.

d0b49a1c-0574-4062-bd68-49a0b4442cad

Wolfart, J.

7824cced-2f37-453b-b8b5-29647679aab3

Jones, Gareth, Streit, M. and Wolfart, J.
(2009)
Wilson's map operations on regulat dessins and cyclotomic fields of definition
*Proceedings of the London Mathematical Society*, 100, (2), . (doi:10.1112/plms/pdp033).

## 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.

**PDF JStWoPLMSproofsCorrected.pdf
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## More information

Published date: 25 August 2009

## Identifiers

Local EPrints ID: 156469

URI: http://eprints.soton.ac.uk/id/eprint/156469

ISSN: 0024-6115

PURE UUID: bcb8b584-ffc9-49f1-966f-fff4a699e9c8

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Date deposited: 02 Jun 2010 10:06

Last modified: 18 Jul 2017 12:42

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## Contributors

Author:
M. Streit

Author:
J. Wolfart

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