CRITERIA FOR COMPLEX MULTIPLICATION AND TRANSCENDENCE PROPERTIES OF AUTOMORPHIC-FUNCTIONS

The results of this paper are joint work with Paula Beazley Cohen. Indeed, this paper is closely based on a larger preprint [CSW] written with her. To our regret, she prefers not to join us for this publication, but instead to give elsewhere a different account of this joint work, which will include other material (see also her prepublication [Co1], and [Co2] for a special case). With her kind permission, we make use of a lot of her ideas, remarks and numerous corrections of previous versions of the present article, without mentioning them at each item. The two authors of the present article accept full responsability for any errors in it that may have been overlooked. The second author thanks PROCOPE for its support and the MSRI Berkeley for its hospitality in Spring 1993, when the lion's share of this manuscript was written. The first author thanks the College de France, Paris, for its support and hospitality in the winter of 1993. The authors were also helped by the CNRS, the DFG and the Centre d'Etudes de Saclay. We are most grateful to URA 763 CNRS, the research team ''Problemes Diophantiens'' at Paris, in particular to Michel Waldschmidt and Daniel Bertrand, for the discussions arising during our participation in its seminar. Finally, the referee of this paper contributed valuable remarks and corrections.