Mathematics

I pursue research in mathematics in the Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, The University of Osaka. This page contains materials related to my research.

Papers

Research on Dessins d’Enfants

At the heart of the theory of dessins d’enfants lies Belyi’s theorem. It follows from this theorem that, for any compact Riemann surface $X$ defined over a number field, there exists a rational function $\beta\colon X \to \mathbb{P}_{\mathbb{C}}^1$ defined over a number field such that the critical values of $\beta$ (those $\beta=c$ for which the equation has a multiple solution on $X$, where $c \in \mathbb{P}_{\mathbb{C}}^1$) are contained in the three-point set $0, 1, \infty$. We call $\beta$ a Belyi function and $(X, \beta)$ a Belyi pair.
For a Belyi pair, one can draw a connected bipartite graph called a dessin d’enfant (French for “child’s drawing”). Here, a bipartite graph means a graph whose vertices are colored black and white and in which every edge joins a black vertex to a white vertex. A dessin d’enfant is embedded on an orientable closed surface of the same genus as $X$—a sphere or a torus with one or more handles—with $\beta^{-1}(0)$ corresponding to the black vertices, $\beta^{-1}(1)$ to the white vertices, and $\beta^{-1}(\infty)$ to the faces. Each face is homeomorphic to a disk.
I study properties of dessins d’enfants and their relationships with algebraic curves.

  • Regularity and Automorphism Groups of Dessins d’Enfants with Uniform Passports (Tatsuya Ohnishi, 2026.2)
    Latest version arXiv(Abstract PDF HTML) Master’s thesis
    We investigated how regularity and automorphism groups are distributed among dessins d’enfants with uniform passports—that is, dessins for which the ramification indices at the black vertices, white vertices, and faces are constant within each class—and how these distributions vary with genus. The focus was on the fact that uniform dessins exhibit a high degree of symmetry but are not necessarily regular. The main results are as follows.
    – A passport of the form $[a^{p}, b^{q}, n]$ (the tree case) admits a regular dessin if and only if $\gcd(p,q)=1$.
    – A passport of the form $[n, b^{q}, n]$ of genus at least two admits a dessin with a trivial automorphism group.
    In addition, I proved several propositions concerning uniform passports of genera 0 and 1. I also proved two theorems on counting elements of symmetric groups that are useful for studying automorphism groups.
    Note: The master’s thesis presents the same results as Version 1 of the paper published on arXiv. The basic material in Chapters 1 and 2 is explained in slightly greater detail, while the content from Chapter 3 onward is identical.
  • Automorphism Groups of Uniform Dessins d’Enfants of Genus at Least Two (Tatsuya Ohnishi, 2026.6)
    Latest version arXiv(Abstract PDF HTML)
    In addition to the results of the previous paper, I proved that every passport of the form $[b^q, b^q, n] (q \ge 2)$ of genus at least two admits a dessin with a trivial automorphism group.
    Furthermore, as counterexamples to a conjecture proposed in the previous paper, I demonstrated the existence of a uniform passport of genus 2 that admits no dessin with a trivial automorphism group and, conversely, the existence of a uniform passport of genus 2 for which every dessin has a trivial automorphism group.
    I also gave an alternative proof of the theorem concerning passports of the form $[n, b^{q}, n]$ established in the previous paper, using the centralizer-counting method developed in this paper.
  • Uniform Unicellular Dessins d’Enfants with Trivial Automorphism Groups (Tatsuya Ohnishi, 2026.9)
    arXiv(Abstract PDF HTML)
    In addition to the results of the previous two papers, I proved that every passport of the form $[a^p, b^q, n] (2 \le p < q)$ of genus at least two admits a dessin with a trivial automorphism group. Consequently, every uniform unicellular passport of genus at least two, as well as every permutation of such a passport, admits a dessin with a trivial automorphism group.

Presentations

Awards