Calculating Complete Lists of Belyi Pairs

Nikolai M. Adrianov and George B. Shabat
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Nikolai M. Adrianov: Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
George B. Shabat: Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia

Mathematics, 2022, vol. 10, issue 2, 1-25

Abstract: Belyi pairs constitute an important element of the program developed by Alexander Grothendieck in 1972–1984. This program related seemingly distant domains of mathematics; in the case of Belyi pairs, such domains are two-dimensional combinatorial topology and one-dimensional arithmetic geometry. The paper contains an account of some computer-assisted calculations of Belyi pairs with fixed discrete invariants. We present three complete lists of polynomial-like Belyi pairs: (1) of genus 2 and (minimal possible) degree 5; (2) clean ones of genus 1 and degree 8; and (3) clean ones of genus 2 and degree 8. The explanation of some phenomena we encounter in these calculations will hopefully stimulate further development of the dessins d’enfants theory.

Keywords: Belyi pairs; dessins d’enfants; algebraic curves; graphs; surfaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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