Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
Mohamed Mahmoud Chems-Eddin
TL;DR
This work investigates Greenberg's conjecture for real biquadratic fields by examining when the 2-Iwasawa module $A(k_\infty)$ has the same rank as the 2-class group at the first cyclotomic layer $A(k_1)$. Using a blend of class-field theory, multiquadratic class-number formulas, unit-index analysis, and ramification data, the paper derives explicit criteria and a finite list of field forms (A–F) for which $\operatorname{rank}(A(K_\infty))\le 2$ and equals $\operatorname{rank}(A(K))$, with detailed cases yielding $A(K_\infty)=0$, $\mathbb{Z}/2^{n}\mathbb{Z}$, or $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2^{n}\mathbb{Z}$. It also presents a constructive framework based on Kuroda's formula and Wada's unit theory to compute the full $2$-Iwasawa module and invariants, applying Fukuda's stabilization results to identify infinite families with abelian Hilbert 2-class towers. The results advance understanding of Greenberg-type behavior in real biquadratic and related triquadratic contexts and provide practical tools for computing $A(k_\infty)$ in concrete settings.
Abstract
The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem: What are real biquadratic number fields $k$ such that ${\rm rank}(A(k_\infty)) = {\rm rank}(A(k_1))$?, where $A(k_\infty)$ is the $2$-Iwasawa module of $k$ and $A(k_1)$ is the $2$-class group of $k_1$ the first layer of the cyclotomic $\mathbb Z_2$-extension of $k$. Moreover, we give several families of real biquadratic fields $k$ such that $A(k_\infty)$ is trivial or isomorphic to $\mathbb Z/2^{n} \mathbb Z$ or $\mathbb Z/2\mathbb Z \times\mathbb Z/2^n \mathbb Z$, where $n$ is a given positive integer. The reader can also find some results concerning the $2$-rank of the class group of certain real triquadratic fields.