Asymptotic growth patterns for class field towers

Abstract

Let $p$ be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a ${\mathbb{Z}}_p$-tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain ${\mathbb{Z}}_p$-extensions in which a given prime above $p$ is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for $p$-primary class numbers of the number fields in a ${\mathbb{Z}}_p$-tower.