Belyi Function Decompositions for The Icosahedron of Genus 4

Natalia Amburg; Mariia Kovaleva

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arXiv · 2405.07092

Belyi Function Decompositions for The Icosahedron of Genus 4

Natalia Amburg·Mariia Kovaleva

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Abstract

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 $\beta_{I_0}: \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $\beta_{I_4}: \mathcal{B} \rightarrow \mathbb{CP}^1$ 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.

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Natalia Amburg, Mariia Kovaleva. 2024-05-11. Belyi Function Decompositions for The Icosahedron of Genus 4. https://arxiv.org/abs/2405.07092

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