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Calculating Complete Lists of Belyi Pairs

Author

Listed:
  • 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
    Department of Mathematics, Logic and Intelligent Systems in Humanities, Russian State University for the Humanities, 125993 Moscow, Russia)

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.

Suggested Citation

  • Nikolai M. Adrianov & George B. Shabat, 2022. "Calculating Complete Lists of Belyi Pairs," Mathematics, MDPI, vol. 10(2), pages 1-25, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:258-:d:725384
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