$\mathbb{Z}_2$-extension of real quadratic fields with $\mathbb{Z}/2\mathbb{Z}$ as $2$-class group at each layer

TL;DR

This paper investigates the structure of 2-class groups in the $bZ_2$-extension of certain real quadratic fields, confirming Greenberg's conjecture for infinitely many such fields under specific conditions.

Contribution

It demonstrates that for real quadratic fields with three prime factors, the 2-class group remains cyclic of order 2 at each layer of the $bZ_2$-extension, validating a case of Greenberg's conjecture.

Findings

2-class group is $bZ/2bZ$ at each layer

Validates Greenberg's conjecture for new family of fields

Infinitely many fields satisfy the conditions

Abstract

Let $K= \mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d$ having three distinct prime factors. We show that the $2$-class group of each layer in the $\mathbb{Z}_2$-extension of $K$ is $\mathbb{Z}/2\mathbb{Z}$ under certain elementary assumptions on the prime factors of $d$. In particular, it validates Greenberg's conjecture on the vanishing of the Iwasawa $\lambda$-invariant for a new family of infinitely many real quadratic fields.