Application of the André–Oort Conjecture to some Questions in Transcendence

Abstract

Abstract We show how a problem concerning the transcendence of values of the classical hypergeometric function, and originating in work of Siegel on G -functions, can be solved using a special case of a conjecture of Andre–Oort on the distribution of complex multiplication (or special) points on algebraic curves in Shimura varieties. The special case in question has recently been proven, at our suggestion, by Edixhoven & Yafaev (2001); see also Yafaev (2001b). This settles the question of which classical hypergeometric functions with rational parameters, satisfying certain natural assumptions, take only finitely many algebraic values at algebraic points. The fact that such a function cannot have an arithmetic monodromy group goes back to work of Wolfart (1988). We introduce a number of related problems. Note added in revision In the original version of this article, we introduced a number of open problems motivated by transcendence questions on the classical hypergeometric function. These are summarised in Problems 1, 2, 3 and 4 of §1. One of the main points of this article is to show how Problems 1 and 2 follow from Problem 4, which is in turn related to the Andre–Oort Conjecture, Oort (1997) concerning the distribution of complex multiplication points on subvarieties of Shimura varieties.