Some applications of the $p$-adic analytic subgroup theorem

We use a $p$-adic analogue of the W\"ustholz' analytic subgroup theorem 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 W\"ustholz contained in [G. W\"ustholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkh\"auser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes \`a un param\`etre $p$-adique de vari\'et\'es de groupe, Invent. Math. 40 (1977), no. 2, 171-193] and [D. Bertrand, Probl\`emes locaux, Soci\'et\'e Math\'ematique de France, Ast\'erisque 60-70 (1979), 163-189].