Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields

TL;DR

This paper investigates the Iwasawa module of cyclotomic $bZ_2$-extensions of real quadratic fields, providing new examples where Greenberg's conjecture holds and methods to compute the module's order.

Contribution

It offers new instances confirming Greenberg's conjecture for specific real quadratic fields and introduces a way to calculate the Iwasawa module's order using units in biquadratic fields.

Findings

Greenberg's conjecture holds for new real quadratic fields with cyclic Iwasawa modules.

A fundamental system of units is constructed for certain biquadratic fields.

The order of the Iwasawa module is computed explicitly in these cases.

Abstract

For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when $X_{\infty}$ is cyclic and the prime is $p=2$. Furthermore, we find a fundamental system of units for certain biquadratic fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{D})$ and show how to use it to calculate the order of $X_{\infty}$.