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.
Highlights
By definition, a Belyi pair is a pair (X, β), where X is a complete smooth irreducible curve over an algebraically closed field k and β is a rational non-constant function on X with only three critical values
We denote the Belyi pairs corresponding to pastings by the Gaussian words defining pastings
Theorem 2. (i) The Belyi pairs (X, β)abcdabcd and (X, β)ababcdcd are defined over Q; √(ii) The remaining two, (X, β)abacbdcd and (X, β)abacdbcd, are Galois-conjugated over some
Summary
A Belyi pair is a pair (X, β), where X is a complete smooth irreducible curve over an algebraically closed field k and β is a rational non-constant function on X with only three critical values This short definition covers some enigmatic relations between several domains of mathematics. According to Belyi’s theorem, the (appropriately defined) category of dessins d’enfants is equivalent to the category of Belyi pairs over C (which is more or less clear from the above explanations) but to such a category over the field Q of algebraic numbers! The theory of dessins d’enfants provides the unique opportunity of the visualization of the absolute Galois group This is just one of the many consequences of the above category equivalence, but it is one of the main motivations of the calculations presented below.
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