Regularity and Automorphism Groups of Dessins d'Enfants with Uniform Passports

Abstract

For a smooth algebraic curve defined over a number field, one can associate a bipartite graph known as a dessin d'enfant. In this paper, we investigate the regularity and automorphism groups of dessins d'enfants with uniform passports, that is, those for which the valencies of black vertices, white vertices, and faces are constant, and study how these properties depend on the genus. Although uniformity imposes a high degree of symmetry, such dessins are not necessarily regular. Our main results are as follows: (1) A passport of the form $[a^{p}, b^{q}, n]$ (the tree case) admits a regular dessin if and only if $\gcd(p,q)=1$. (2) Every passport of the form $[n, b^{q}, n]$ of genus at least 2 admits a dessin with a trivial automorphism group. In addition, we obtain several results on uniform passports of genus 0 and 1. We also establish two theorems on the enumeration of elements in symmetric groups, which are useful for the study of automorphism groups of dessins.