The class group of z_p-extensions over totally real number fields

Let p be an odd prime. We shall give a criterion for p-divisibility of the class number of the w-th layer of a Z_p-extension over a certain totally real number field by means of the value of the p-adic zeta function. We shall also discuss the capitulation in such a Z_p-extension and a sufficient condition for the Iwasawa λ andπ-invariants of it to vanish.