Belyi functions for hyperbolic hypergeometric-to-Heun transformations

TL;DR

This paper classifies all Belyi functions that transform specific hypergeometric equations into Heun equations, revealing 366 Galois orbits and developing new algorithms to compute these functions despite their high degree.

Contribution

It provides a complete classification of Belyi functions for hypergeometric-to-Heun transformations with hyperbolic parameters, including the development of efficient algorithms for their computation.

Findings

366 Galois orbits of Belyi functions identified

Maximum degree of these functions is 60

Two new algorithms developed for computing Belyi functions

Abstract

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k,1/l,1/m that satisfy k,l,m in N and the hyperbolic condition 1/k+1/l+1/m<1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.