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An explicit quasiplatonic curve with non-abelian moduli field

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Abstract

We give an example of a regular dessin d’enfant whose field of moduli is not an abelian extension of the rational numbers, namely it is the field generated by a cubic root of 2. This answers a previous question. We also prove that the underlying curve has non-abelian field of moduli itself, giving an explicit example of a quasiplatonic curve with non-abelian field of moduli. In the last section, we note that two examples in previous literature can be used to find other examples of regular dessins d’enfants with non-abelian field of moduli.

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Notes

  1. If \(H<G\), we define \(\mathrm {core}_GH=\bigcap _{g\in G} gHg^{-1}\).

  2. From now on, when a, b are elements of a group, we denote \(a^b=b^{-1}ab\).

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Acknowledgments

I would like to thank my master’s thesis advisor, Andrei Jaikin, for an immense amount of help and encouragement while I was working on this problem. I would also like to thank the reviewers of this article for many insightful comments which have lead to substantial improvements.

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  1. Moisés Herradón Cueto

    Present address: Department of Mathematics, University of Wisconsin, Van Vleck Hall, 480 Lincoln Drive, Madison, WI, 53706, USA

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  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, C/ Francisco Tomás y Valiente, 7, 28049, Madrid, Spain

    Moisés Herradón Cueto

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  1. Moisés Herradón Cueto
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Correspondence to Moisés Herradón Cueto.

Additional information

Supported by the program Posgrado de Excelencia Internacional en Matemáticas 2013-2014 at Universidad Autónoma de Madrid.

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Herradón Cueto, M. An explicit quasiplatonic curve with non-abelian moduli field. Rev Mat Complut 29, 725–739 (2016). https://doi.org/10.1007/s13163-016-0196-z

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