Greenberg's conjecture for real quadratic fields and the cyclotomic $\mathbb{Z}_2$-extensions

TL;DR

This paper investigates Greenberg's conjecture for real quadratic fields by analyzing the 2-part of class groups in cyclotomic extensions, providing a method to study unit indices, and confirming the conjecture for fields with discriminant less than 10000.

Contribution

It introduces a new method to analyze the index of cyclotomic units and proves Greenberg's conjecture for all real quadratic fields with discriminant under 10000.

Findings

The class group sequence stabilizes for fields with discriminant less than 10000.

Greenberg's conjecture holds for all such real quadratic fields.

A new approach to study the index of cyclotomic units in unit groups.

Abstract

Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.