A classification of coverings yielding Heun-to-hypergeometric reductions

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 $\mathbb{P}^{1}\to\mathbb{P}^{1}$ that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases with Liouvillian solutions). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 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). 38 of these coverings appear in Herfurtner's list of elliptic surfaces over $\mathbb{P}^{1}$ with four singular fibers, as their $j$-invariants. In passing, we show in an elegant way that there are no coverings with some branching patterns.