Fuchsian Equations with Three Non-Apparent Singularities

Abstract

We show that for every second-order Fuchsian linear differential equation E with n singularities, of which n−3 are apparent, there exists a hypergeometric equation H and a linear differential operator with polynomial coefficients which maps the space of solutions of H into the space of solutions of E. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations E with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature 1 on the punctured sphere with conic singularities, all but three of them having integer angles.