A classification of coverings yielding Heun-to-hypergeometric reductions

TL;DR

This paper classifies specific coverings that enable transformations from hypergeometric to Heun equations, identifying 61 parametric transformations and their underlying Belyi coverings, enriching the understanding of these special functions.

Contribution

It provides a comprehensive classification of coverings leading to hypergeometric-to-Heun transformations, including explicit descriptions and connections to elliptic surfaces.

Findings

61 hypergeometric-to-Heun transformations identified

48 Belyi coverings realize these transformations

28 transformations are compositions of smaller degree maps

Abstract

Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases when the involved hypergeometric equation has cyclic or dihedral monodromy). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 pull-backs are compositions of smaller degree transformations between hypergeometric and Heun functions. The 61 transformations are realized by 48 different Belyi coverings (though 2 coverings should be counted twice as their moduli field is quadratic). The same Belyi coverings appear in several other contexts. For example, 38 of the coverings appear in Herfutner's list of elliptic surfaces over P1 with four singular fibers, as their j-invariants. In passing, we demonstrate an elegant way to show that there are no coverings P1-to-P1 with some branching patterns.