On the failure of pseudo-nullity of Iwasawa modules

Abstract

Consider the family of CM-fields which are pro- p p p p -adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z p \mathbf {Z}_p -extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro- p p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p p -adic Lie groups. The proof uses Kida’s formula for the growth of λ \lambda -invariants in cyclotomic Z p {\mathbf Z}_p -extensions of CM-fields. In fact, we give a new proof of Kida’s formula which includes a slight weakening of the usual μ = 0 \mu = 0 assumption. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.