Abstract
We consider a non-algebraic Schwarz map D for the hypergeometric differential equation of Appell-Lauricella type with rather general rational parameters. Under which conditions can D take algebraic values at algebraic vector arguments τ ? As a necessary condition, the canonical Prym variety lying in the Jacobian of the hypergeometric curve belonging to τ must have a factor with complex multiplication. However, it will be shown that in a Zariski dense subset of arguments few contiguous Schwarz maps D can take algebraic values D(τ) even if this CM condition is satisfied. Explicit examples show that there are exceptional arguments where this statement fails.
