Some applications of the p-adic analytic subgroup theorem

Abstract

We use a $p$-adic analogue of the analytic subgroup theorem of Wustholz to deduce the transcendence and linear independence of some new classes of $p$-adic numbers. In particular we give $p$-adic analogues of results of Wustholz contained in [G. Wustholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkhauser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes a un parametre $p$-adique de varietes de groupe, Invent. Math. 40 (1977), no. 2, 171-193] and [D. Bertrand, Problemes locaux, Societe Mathematique de France, Asterisque 60-70 (1979), 163-189].