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Genus one Belyi maps by quadratic correspondences

Raimundas Vidunas

Institute of Applied Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

E-mail Address: [email protected]

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and

Yang-Hui He

http://orcid.org/0000-0002-0787-8380

School of Physics, NanKai University, Tianjin 300071, P. R. China

Department of Mathematics, City, University of London, EC1V 0HB, UK

Merton College, University of Oxford, OX14JD, UK

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S0129167X20500342Cited by:0 (Source: Crossref)

Abstract

We present a method of obtaining a Belyi map on an elliptic curve from that on the Riemann sphere. This is done by writing the former as a radical of the latter, which we call a quadratic correspondence, with the radical determining the elliptic curve. With a host of examples of various degrees, we demonstrate that the correspondence is an efficient way of obtaining genus one Belyi maps. As applications, we find the Belyi maps for the dessins d’enfant which have arisen as brane-tilings in the physics community, including ones, such as the so-called suspended pinched point, which have been a standing challenge for a number of years.

Keywords:

AMSC: 11G32, 14H52, 81T30

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