Belyi function decompositions for the icosahedron of genus~$4$

N. Ya. Amburg ; M. A. Kovaleva

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 3-30

Résumé

The icosahedron $I_4$ of genus $4$ is a dessin d'enfant embedded in Bring's curve $\mathcal{B}$. The dessin $I_4$ is related in some sense to a regular icosahedron $I_0$ embedded in the complex Riemann sphere. In particular, decompositions of Belyi functions $\beta_{I_0}\colon \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $\beta_{I_4}\colon \mathcal{B} \rightarrow \mathbb{CP}^1$ for $I_0$ and $I_4$ have the same lattice. The diagram of $\beta_{I_0}$ decompositions is already known. In the present paper we find $\beta_{I_4}$ decompositions. Note that $\beta_{I_0}$ decomposes into rational functions on $\mathbb{C}P^1$, while in case of $\beta_{I_4}$ we deal with maps between different algebraic curves.