Darboux evaluations for hypergeometric functions with the projective monodromy PSL(2,F7)

Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already been considered for algebraic F-2(1)-functions. This article presents Darboux evaluations for the classical case of F-3(2)-functions with the projective monodromy group PSL(2, F-7). The pullback curves are of genus 0 (in the simplest case) or of genus 1. As an application of the genus 0 evaluations, appealing modular evaluations of the same F-3(2)-functions are derived.