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Counting Belyi Pairs over Finite Fields

Part of the MATRIX Book Series book series (MXBS,volume 1)

Abstract

Alexander Grothendieck’s theory of dessins d’enfants relates Belyi pairs over \(\overline {\mathbb {Q}}\) with certain graphs on compact oriented surfaces; the present paper is aimed at the extension of this correspondence. We introduce two closely related categories of Belyi pairs over arbitrary algebraically closed fields, in particular over the algebraic closures \(\overline {\mathbb {F}_p}\) of finite fields. The lack of the analogs of graphs on surfaces over \(\overline {\mathbb {F}_p}\) promotes the development of other tools that are introduced and discussed. The problem of counting Belyi pairs of bounded complexity is posed and illustrated by some examples; the application of powerful methods of counting dessins d’enfants together with the concept of bad primes is emphasized. The relations with geometry of the moduli spaces of curves is briefly mentioned.

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Acknowledgements

The paper is supported in part by the Simons foundation.

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Correspondence to George Shabat .

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Shabat, G. (2018). Counting Belyi Pairs over Finite Fields. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_16

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