arXiv subjects

Madoka Horie

Publications and source records attributed to Madoka Horie.

4 recordsLinked to original sources

arXiv2026

An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup

In this paper, we explicitly determine an algebraic Belyi function on a unique smooth projective model $\widetilde{W}$ of the Wiman sextic curve $W$ and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup $Γ_{\widetilde{W}} \subset {\rm SL}_2(\mathbb{Z})$. As an application, we give a direct proof of the unbounded denominators conjecture in weight $2$ for $Γ_{\widetilde{W}}$. The conjecture is now known in full generality by the work of Calegari, Dimitrov, and Tang, following earlier progress including work of Dong, Lin, and Ng. Our proof, however, uses a degeneration of $\widetilde{W}$ over $\mathbb{F}_{5}$ together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.

math.NT↗

arXiv2026

The $L$-function of the surface parametrizing cuboids

In this note, we compute the $L$-function of the projective smooth surface $S$ over $\mathbb{Q}$ that parametrizes cuboids whose geometric properties are studied in detail by Stoll and Testa. As a byproduct, we completely determine the structure of ${\rm Pic}(S_{\overline{\mathbb{Q}}})$ as a ${\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-module.

math.NT↗

arXiv2025

Computing algebraic Belyi functions on Bring's curve

In this paper, we explicitly compute two kinds of algebraic Belyi functions on Bring's curve. One is related to a congruence subgroup of ${\rm SL}_2(\mathbb{Z})$ and the other is related to a congruence subgroup of the triangle group $Δ(2,4,5)\subset \SL_2(\R)$. To carry out the computation, we use elliptic cusp forms of weight 2 for the former case and the automorphism group of Bring's curve for the latter case. We also discuss a suitable base field (a number field) for describing isomorphisms between Hulek-Craig's curve, Bring's curve, and another algebraic model obtained as a modular curve.

math.NT↗

arXiv2022

Equivalence classes of dessins d'enfants with two vertices

Let $N$ be a positive integer. For any positive integer $L\leq N$ and any positive divisor $r$ of $N$, we enumerate the equivalence classes of dessins d'enfants with $N$ edges, $L$ faces and two vertices whose automorphism groups are cyclic of order $r$. Further, for any non-negative integer $h$, we enumerate the equivalence classes of dessins with $N$ edges, $h$ faces of degree $2$ with $h\leq N$, and two vertices, whose automorphism groups are cyclic of order $r$. Our arguments are essentially based upon a natural one-to-one correspondence of the equivalence classes of all dessins with $N$ edges to the equivalence classes of all pairs of permutations with components generating transitive subgroups of the symmetric group of degree $N$.

math.NT↗