Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Tatsuya Ohnishi

TL;DR

This work addresses the symmetry of dessins d'enfants arising from uniform passports across genus. It combines Belyi theory with detailed symmetric-group analysis, leveraging cycle-type counts $T(b,q)$, $N(b,q)$ and $I_m(b,q)$ and triangle-group methods to relate regularity to monodromy primitivity. The paper proves two key results: (i) for tree-type uniform passports, a regular dessin exists if and only if gcd$(p,q)=1$, and analogous permutations; (ii) for genus $\ge 2$, uniform passports of the form $[n,b^{q},n]$ always admit a dessin with trivial automorphism group, with parallel results for $[b^{q},n,n]$ and $[n,n,b^{q}]$. It also provides genus-0 and genus-1 classifications of uniform dessins and presents two symmetric-group enumeration theorems that aid automorphism-group analysis. Together, these results illuminate how symmetry in uniform dessins evolves with genus and contribute to understanding fields of moduli versus fields of definition in the descent problem.

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.

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

TL;DR

This work addresses the symmetry of dessins d'enfants arising from uniform passports across genus. It combines Belyi theory with detailed symmetric-group analysis, leveraging cycle-type counts , and and triangle-group methods to relate regularity to monodromy primitivity. The paper proves two key results: (i) for tree-type uniform passports, a regular dessin exists if and only if gcd, and analogous permutations; (ii) for genus , uniform passports of the form always admit a dessin with trivial automorphism group, with parallel results for and . It also provides genus-0 and genus-1 classifications of uniform dessins and presents two symmetric-group enumeration theorems that aid automorphism-group analysis. Together, these results illuminate how symmetry in uniform dessins evolves with genus and contribute to understanding fields of moduli versus fields of definition in the descent problem.

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 (the tree case) admits a regular dessin if and only if . (2) Every passport of the form 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.
Paper Structure (22 sections, 21 theorems, 270 equations, 9 figures, 4 tables)

This paper contains 22 sections, 21 theorems, 270 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Belyı̆'s Theorem and Dessins d'Enfants
  3. Regularity and Automorphism Groups
  4. Uniform Passports and Dessins
  5. Related Work
  6. Purpose and Main Results
  7. Preliminaries
  8. Estimates for the Symmetric Group
  9. Elements of Cycle Type $(b^{q})$
  10. Lower Bound for $N/T$
  11. Computation of $I_{m}$
  12. Genus 0
  13. Genus 1
  14. Regularity Criteria in Arbitrary Genus
  15. Tree Cases
  16. ...and 7 more sections

Key Result

Theorem 1.1

Belyi79Belyi02Jones16 Let $X$ be a compact Riemann surface, that is, a smooth projective algebraic curve in the complex projective space $\mathbb{P}_{\mathbb{C}}^{N}$ for some $N$. Then $X$ can be defined over the field of algebraic numbers $\overline{\mathbb{Q}}$ if and only if there exists a non-c

Figures (9)

  • Figure 1: Examples of dessins d'enfants
  • Figure 2: Regular and non-regular dessins with the same passport
  • Figure 3: Uniform dessins d'enfants of genus 0, degree 6
  • Figure 4: Uniform dessins d'enfants of genus 1, degree 6
  • Figure 5: Uniform dessins d'enfants of genus 2, degree 6
  • ...and 4 more figures

Theorems & Definitions (52)

  • Theorem 1.1: Belyı̆'s Theorem
  • Definition 2.1: Dessin d'enfant
  • Definition 2.2: Passport of a dessin
  • Definition 2.3: Uniform passports and dessins
  • Definition 2.4: Monodromy group of a dessin d'enfant
  • Remark 2.5
  • Definition 2.6: Regular dessins
  • Lemma 2.7
  • proof
  • Definition 2.8: Automorphism group of a dessin d'enfant
  • ...and 42 more