On the structure and stability of ranks of $2$-class groups in cyclotomic $\mathbb{Z}_{2}$-extensions of certain real quadratic fields

TL;DR

This paper investigates the structure and stability of 2-class groups in the cyclotomic Z_2-extension of certain real quadratic fields, providing new characterizations and conditions for their isomorphism types and confirming the vanishing of the Iwasawa mu-invariant.

Contribution

It offers new criteria for the 2-class group structure in real quadratic fields with four prime factors, extending previous results and analyzing the stability of class group ranks in the cyclotomic extension.

Findings

Class groups stabilize in rank across layers of the extension.

Conditions identified for specific isomorphism types of class groups.

Alternate proof for the vanishing of the Iwasawa mu-invariant.

Abstract

For a real quadratic field $K= \mathbb{Q}(\sqrt{d})$ with discriminant $D_{K}$ having four distinct prime factors, we study the structure of the $2$-class group $A(K_{1})$ of the first layer $K_{1} = \mathbb{Q}(\sqrt{2},\sqrt{d})$ of the cyclotomic $\mathbb{Z}_{2}$-extension of $K$. With some suitably convenient assumptions on the rank and the order of $A(K_{1})$, we characterize $K$ for which the $2$-class group $A(K)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. We infer that the $2$-ranks of the class groups in each layer stabilizes by virtue of a result of Fukuda. This also provides an alternate way to establish that the Iwasawa $\mu$-invariant of $K$ vanishes. In some cases, we also provide sufficient conditions on the constituent prime factors of $D_{K}$ that imply $A(K) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, $A(K_{1}) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}$ and $A(K^{\prime}) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, where $K^{\prime} = \mathbb{Q}(\sqrt{2d})$. This extends some results obtained by Mizusawa.