Introduction

Effective results in the diophantine approximation of algebraic numbers are difficult to obtain, and for a long time the only general method available was Baker's theory of linear forms in logarithms. An alternative, more algebraic, method was later proposed in Bombieri and Bombieri and Cohen. This new method is quite different from the classical approach through the theory of linear forms in logarithms.

In this paper, we improve on results derived in. These results concern effective approximations to roots of high order of algebraic numbers and their application to diophantine approximation in a number field by a finitely generated multiplicative subgroup. We restrict our attention to the nonarchimedean case, although our results and methods should go over mutatis mutandis to the archimedean setting.

We do not claim that our theorems are the best that are known in this direction. Linear forms in two logarithms (which are easier to treat than the general case) suffice to prove somewhat better results than our Theorem 5.1, see Bugeaud and Bugeaud and Laurent; we give an explicit comparison in §5, Remark 5.1.

Theorem 5.2, which is useful for general applications, follows from Theorem 5.1 by means of a trick introduced for the first time in and improved in. Thus any improved form of Theorem 5.1 carries automatically an improvement of Theorem 5.2.

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 André–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 André–Oort Conjecture, Oort (1997) concerning the distribution of complex multiplication points on subvarieties of Shimura varieties.