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We consider Schwarz maps for triangles whose angles are rather general rational multiples of . Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.

This text is based on lectures by the author in the Summer School in Istanbul in June 2005. It gives a review of some of the basic aspects of the theory of hypergeometric structures of Gelfand, Kapranov and Zelevinsky, including Differential Equations, Integrals and Series, with emphasis on the latter. The Secondary Fan is constructed and subsequently used to describe the ‘geography’ of the domains of convergence of the Γ-series. A solution to certain Resonance Problems is presented and applied in the context of Mirror Symmetry. Many examples and some exercises are given throughout the paper.

This is an introduction to complex orbifolds with an emphasis on orbifolds in dimension 2 and covering relations between them.

The hypergeometric function is a slight generalization of the power fucntion. We will see this by the Schwarz map of the hypergeometric equation focussing on the behavior of this map when the local exponent-differences are purely imaginary

This article contains the open problems discussed during the problem session of the CIMPA summer school “Arithmetic and Geometry Around Hypergeometric Functions” held at Galatasaray University, İstanbul, 2005.