DARBOUX EVALUATIONS OF ALGEBRAIC GAUSS HYPERGEOMETRIC FUNCTIONS

Algebraic Gauss hypergeometric functions can be expressed explicitly in several ways. One attractive way is to pull-back their hypergeometric equations (with a finite monodromy) to Fuchsian equations with a finite cyclic monodromy, and express the algebraic solutions as radical functions on the covering curve. This article presents these pull-back transformations of minimal degree for the hypergeometric equations with the tetrahedral, octahedral or icosahedral projective monodromy. The minimal degree is 4, 6 or 12, respectively. The covering curves are called Darboux curves, and they have genus zero or (for some icosahedral Schwarz types) genus one.