Class number behavior in a two-tiered tower of $\mathbb{Z}_p$-extensions

TL;DR

This paper investigates the growth of the p-part of class numbers and the structure of unramified Iwasawa modules in a two-tiered tower of $ ext{Z}_p$-extensions over number fields, revealing new structural insights.

Contribution

It provides new results on class number behavior and Iwasawa module structure in a layered $ ext{Z}_p$-extension setting, extending previous Iwasawa theory analyses.

Findings

Characterization of class number growth in intermediate fields

Analysis of unramified Iwasawa modules in specific cases

Structural properties of $ ext{Z}_p$-extension towers

Abstract

Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension field of an algebraic number field $k$. Moreover, we take a $\mathbb{Z}_p$-extension $K_\infty$ over $k_\infty$. In this paper, we study the behavior of the $p$-part of the class number of certain intermediate fields of $K_\infty /k$. We also consider the structure of the unramified Iwasawa module of $K_\infty / k_\infty$ for several cases.