Rarity of pseudo-null Iwasawa modules for p-adic Lie extensions

In this paper, we obtain a necessary and sufficient condition for the pseudo-nullity of the p-ramified Iwasawa module for p-adic Lie extension of totally real fields. It is applied to answer the corresponding question for the minus component of the unramified Iwasawa module for CM-fields. The results show that the pseudo-nullity is very rare.