Almost Abelian regular dessins d'enfants

Ruben A. Hidalgo¹

¹ Departamento de Matemática Universidad Técnica Federico Santa María Casilla 110-V Valparaíso, Chile

Fundamenta Mathematicae, Tome 222 (2013) no. 3, pp. 269-278.

Résumé

A regular dessin d'enfant, in this paper, will be a pair $(S, \beta)$, where $S$ is a closed Riemann surface and $\beta:S \to \widehat{\mathbb C}$ is a regular branched cover whose branch values are contained in the set $\{\infty,0,1\}$. Let ${\rm Aut}(S,\beta)$ be the group of automorphisms of $(S,\beta)$, that is, the deck group of $\beta$. If ${\rm Aut}(S,\beta)$ is Abelian, then it is known that $(S,\beta)$ can be defined over ${\mathbb Q}$. We prove that, if $A$ is an Abelian group and ${\rm Aut}(S,\beta) \cong A \rtimes {\mathbb Z}_{2}$, then $(S,\beta)$ is also definable over ${\mathbb Q}$. Moreover, if $A \cong {\mathbb Z}_{n}$, then we provide explicitly these dessins over ${\mathbb Q}$.

Détail

DOI : 10.4064/fm222-3-3

Keywords: regular dessin denfant paper pair beta where closed riemann surface beta widehat mathbb regular branched cover whose branch values contained set infty aut beta group automorphisms beta deck group beta aut beta abelian known beta defined nbsp mathbb prove abelian group aut beta cong rtimes mathbb beta definable mathbb moreover cong mathbb provide explicitly these dessins nbsp mathbb