Calculating Complete Lists of Belyi Pairs

Mathematics

George B. Shabat¹  Nikolai M. Adrianov¹ 

[1] Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia;

关键词: Belyi pairs;    dessins d’enfants;    algebraic curves;    graphs;    surfaces;   

DOI  :  10.3390/math10020258

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown