Valeurs Exceptionnelles de Fonctions Hypergéométriques d’Appell (Exceptional Values of Appell Hypergeometric Functions)

Abstract

In his article [18], J. Wolfart studied the following exceptionnal set ${\cal E}:=\{x\in\bar{\rm Q} : F(a;b;c;x)\in\bar{\rm Q}\}$ where F is the classical, or Gauss hypergeometric function. The first aim of the present article is to describe the exceptional set in the case of Appell hypergeometric functions, which are a generalization to two variables of the Gauss functions. The link will then be made between, on the one hand, the distribution of complex multiplication points (described by Appell function in the article [5] of P. Cohen and J. Wolfart) on a fixed modular variety, using a André-Oort conjecture, and on the other hand, the arithmeticity of the monodromy group related to this function. Lastly, we will see how the localization of certain complex multiplication points leads to the transcendance of the values of Appell hypergeometric functions, at algebraic points.