Belyi functions for hyperbolic hypergeometric-to-Heun transformations
Mark van Hoeij, Raimundas Vidunas
One place where Belyi functions occur is pullback transformations of hypergeometric differential equations to Fuchsian equations with few singularities. This paper presents a complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations (i.e., canonical Fuchsian equations with 4 singularities). 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. In total, we find 872 such Belyi functions up to Mobius transformations, in 366 Galois orbits. 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 hypergeometric-to-Heun transformations.