# Hyberbolic Belyi maps and Shabat-Blaschke products

## TL;DR

This paper introduces hyperbolic analogues of Belyi maps and explores their properties, focusing on Shabat-Blaschke products and their special Chebyshev variants, revealing new arithmetic and functional identities.

## Contribution

It presents the first study of hyperbolic Belyi maps and Shabat-Blaschke products, including Chebyshev-Blaschke products and their arithmetic properties.

## Key findings

- Arithmetic properties of Chebyshev-Blaschke coefficients
- Landen-type identities for theta functions
- Hyperbolic dessins d'enfants in the unit disk

## Abstract

We first introduce hyperbolic analogues of Belyi maps, Shabat polynomials and Grothendieck's dessins d'enfant. In particular we introduce and study the Shabat-Blaschke products and the size of their hyperbolic dessin d'enfants in the unit disk. We then study a special class of Shabat-Blaschke products, namely the Chebyshev-Blaschke products. Inspired by the work of Ismail and Zhang (2007) on the coefficients of the Ramanujan's entire function, we will give similar arithmetic properties of the coefficients of the Chebyshev-Blaschke products and then use them to prove some Landen-type identities for theta functions.