Almost Abelian regular dessins d'enfants

Volume 222 / 2013

Ruben A. Hidalgo Fundamenta Mathematicae 222 (2013), 269-278 MSC: 14H57, 30F10, 05C65. DOI: 10.4064/fm222-3-3


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 ${\mathbb Q}$.


  • Ruben A. HidalgoDepartamento de Matemática
    Universidad Técnica Federico Santa María
    Casilla 110-V
    Valparaíso, Chile

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