The field of moduli and fields of definition of dessins d'enfants

TL;DR

This paper explores dessins d'enfants as topological, algebraic, and combinatorial objects, proves Belyi's theorem linking them to number fields, and investigates their fields of moduli and automorphism properties.

Contribution

It establishes conditions under which dessins d'enfants are defined over their fields of moduli and provides explicit examples with non-abelian field extensions.

Findings

Several dessins are defined over their field of moduli.

Regular dessins with trivial automorphisms are defined over their field of moduli.

Examples of dessins with non-abelian fields of moduli are constructed.

Abstract

We introduce dessins d'enfants from the various existing points of view: As topological covering spaces, as surfaces with triangulations, and as algebraic curves with functions ramified over three points. We prove Belyi's theorem that such curves are defined over number fields, and define the action of the absolute Galois group Gal($\overline Q/Q$) on dessins d'enfants. We prove that several kinds of dessins d'enfants are defined over their field of moduli: regular dessins, dessins with no nontrivial automorphisms and dessins with one face. In the last part, we give two examples of regular dessins d'enfants with a field of moduli that is not an abelian extension of Q. Both of the examples have genus 61 and field of moduli Q(2^(1/3)).