Arithmetic properties of Schwarz maps

The subject of this article belongs to the general question Under which condition(s) suitably normalized transcendental functions take algebraic values at algebraic arguments? Already the classical examples of Weierstrass’ result concerning the exponential function and Theodor Schneider’s result about the elliptic modular function show that arguments and values in these cases are of particular arithmetical interest. Here we try to answer this question for the case of Schwarz maps belonging to Appell–Lauricella hypergeometric functions FD in two and more variables, generalizing our results in [SW2] about Schwarz triangle functions, i.e. for the classical Gauss hypergeometric functions in one variable.