arXiv subjects

Tatsuya Ohnishi

Publications and source records attributed to Tatsuya Ohnishi.

3 recordsLinked to original sources

arXiv2026

Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups

A Bely\uı function on a smooth projective algebraic curve defined over a number field determines a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In previous papers, we proved that every passport of the form $[n,n,n]$, $[n,b^{q},n]$, or $[b^{q},b^{q},n]$ of genus at least $2$ admits a dessin with trivial automorphism group. In this paper, we prove the analogous result for passports of the form $[a^{p},b^{q},n]$. The proof is mainly based on a counting argument: we compare a lower bound for the number of permutation representations having the prescribed passport with an upper bound for the number admitting a nontrivial automorphism. Together with our previous results, this shows that every uniform unicellular passport of genus at least $2$ admits a dessin with trivial automorphism group.

math.AG↗

arXiv2026

Automorphism Groups of Uniform Dessins d'Enfants of Genus at Least Two

For a smooth algebraic curve defined over a number field, one can associate a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In a previous paper, we proved that every passport of the form $[n,b^{q},n]$ of genus at least 2 admits a dessin with trivial automorphism group. Here we prove the analogous result for passports of the form $[b^{q},b^{q},n]$. We also construct examples of uniform passports of genus at least 2 for which every dessin with that passport has nontrivial automorphism group, and others for which every dessin with that passport has trivial automorphism group. Finally, we give an alternative proof of the $[n,b^{q},n]$ case using counting arguments based on centralizers of permutations.

math.AG↗

arXiv2026

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

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.

math.AG↗