Towards a constructive proof of a theorem of Koeck-Lau-Singerman

Details Towards a constructive proof of a theorem of Koeck-Lau-Singerman Towards a constructive proof of a theorem of Koeck-Lau-Singerman

2012-03-28

arxiv.org/abs/1203.6313v1

Mathematics

Complex Variables

Scientific paper

In a couple of recent paper, Koeck-Lau-Singerman have proved that every symmetric Belyi curve $C$ is definable over ${\mathbb R} \cap \bar{\mathbb Q}$. Their proof is based on Weil's Galois descent theorem, so it asserts the existence of some isomorphism $R:C \to Z$, where $Z$ is a curve defined over ${\mathbb R} \cap \bar{\mathbb Q}$. In this paper we work out an alternative proof which provides a method to obtain explicit equations for $R$ and $Z$. In fact, we are able to obtain the following stronger result. If both $C$ and the symmetry are defined over the number field ${\mathbb K}$, then $Z$ is definable over ${\mathbb K} \cap {\mathbb R}$.

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