Abstract: In this paper, we study values at algebraic points of Lauricella hypergeometric functions in n complex variables, n ? 1. We are mostly interested in criteria for the transcendence of these values. The combined results of [P. Cohen and G. W?z, Application of the Andréort conjecture to some questions in transcendence, A panorama of number theory or the view from Baker?s garden (Z? 1999), pp. 89-106, Cambridge Univ. Press, Cambridge, 2002], [Invent. Math. 92 (1988), 187-216] determine necessary and sufficient conditions on the parameters a, b, c for finiteness of the exceptional set of the classical hypergeometric function of one complex variable These results rely on W?z?s analytic subgroup theorem [G. W?z, Algebraic groups, Hodge theory and transcendence, Proc. of the Intern. Congress of Math., Berkeley, California, U.S.A., 1986], [Ann. Math. 129 (1989), 501-517] and on a known particular case, proved in [Ann. Math. 157 (2003), 621-645], of the Andréort conjecture on the distribution of complex multiplication points on curves in Shimura varieties. The results of [P. Cohen and G. W?z, Application of the Andréort conjecture to some questions in transcendence, A panorama of number theory or the view from Baker?s garden (Z? 1999), pp. 89-106, Cambridge Univ. Press, Cambridge, 2002] were generalized to the two (n = 2) variable Appell hypergeometric function by the author [Ramanujan J. 8(3) (2004), 331-355], subject this time to the Andréort conjecture for surfaces in Shimura varieties. In the present paper, we treat the case of several (n ? 3) complex variables. The main contribution of the present paper is the construction for the Lauricella function of the appropriate exceptional set that allows for the application of the Andréort conjecture for n-dimensional subvarieties of Shimura varieties. Some additional results on transcendence of values of Lauricella functions are given, as well as a new counterexample to a conjecture of Coleman. |