Zariski-density of exceptional sets for hypergeometric functions

Abstract

We extend the results of Desrousseaux in [10], [11], [12]. In those papers an exceptional set was constructed for the Appell– Lauricella hypergeometric functions associated to rational ball tuples. The definition of this set requires quite subtle conditions necessary for the application of the Andre–Oort conjecture. In this paper, we show that generalizations of the Andre–Oort conjecture by Pink [20] lead to similar results for a more natural exceptional set, namely the set of algebraic points at which the function takes algebraic values. Desrousseaux’s exceptional set is in general a proper subset of this set.