Asymptotic growth patterns for class field towers

TL;DR

This paper investigates the asymptotic growth patterns of Galois groups over number fields in $Z_p$-extensions, extending classical Iwasawa theory to non-commutative analogues of ray class groups.

Contribution

It establishes precise asymptotic lower bounds for the growth of non-commutative Galois groups in certain $Z_p$-extensions with split primes, advancing understanding of class field tower behaviors.

Findings

Proves asymptotic lower bounds for Galois group growth

Extends classical Iwasawa results to non-commutative settings

Analyzes growth patterns in specific $Z_p$-extensions

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.