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Greenberg's conjecture and Iwasawa module of Real biquadratic fields II

Mohamed Mahmoud Chems-Eddin, Hamza El Mamry

TL;DR

This work analyzes the stability of the $2$-part of the class group in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic fields, proving the main theorem that $\mathrm{rank}(A(K_\infty))\le 2$ and identifying precisely when $\mathrm{rank}(A_\infty(K))=\mathrm{rank}(A(K))$, thereby classifying real biquadratic fields with stable $2$-class groups. It provides a complete list of fields with trivial $2$-Iwasawa module and exhibits infinite families where stability occurs with $\mathrm{rank}(A(K_\infty))=\mathrm{rank}(A(K))=3$, supported by genus-theoretic and unit-theoretic methods (e.g., Wada's construction) combined with Fukuda's stabilization theorem. The paper also extends Greenberg's conjecture to new families, including a detailed treatment of fields of the form $K_\nu=\mathbb{Q}(\sqrt{\nu q},\sqrt{rs})$, establishing many cases where $\mathrm{rank}(A(K_\nu))=\mathrm{rank}(A_\infty(K_\nu))$ and providing explicit residue-symbol criteria. Overall, the results advance understanding of $2$-class groups in cyclotomic towers of real biquadratic fields and contribute new verified instances of Greenberg's conjecture.

Abstract

In this paper we are interested in the stability of the $2$-rank of the class group in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic fields. In fact, we give several families of real biquadratic fields $K$ such that $ rank(A(K)) =rank(A_\infty(K))$ and $rank(A(K))\leq 3$, where $A(K)$ and $A_\infty(K)$ are the $2$-class group and the $2$-Iwasawa module of $K$ respectively. Moreover, Greenberg's conjecture is verified for some new families of number fields; in particular, we determine the complete list of all real biquadratic fields with trivial $2$-Iwasawa module. This work is a continuation of M. M. Chems-Eddin, Greenberg's conjecture and Iwasawa module of real biquadratic fields I, J. Number Theory, 281 (2026), 224-266.

Greenberg's conjecture and Iwasawa module of Real biquadratic fields II

TL;DR

This work analyzes the stability of the -part of the class group in the cyclotomic -extension of real biquadratic fields, proving the main theorem that and identifying precisely when , thereby classifying real biquadratic fields with stable -class groups. It provides a complete list of fields with trivial -Iwasawa module and exhibits infinite families where stability occurs with , supported by genus-theoretic and unit-theoretic methods (e.g., Wada's construction) combined with Fukuda's stabilization theorem. The paper also extends Greenberg's conjecture to new families, including a detailed treatment of fields of the form , establishing many cases where and providing explicit residue-symbol criteria. Overall, the results advance understanding of -class groups in cyclotomic towers of real biquadratic fields and contribute new verified instances of Greenberg's conjecture.

Abstract

In this paper we are interested in the stability of the -rank of the class group in the cyclotomic -extension of real biquadratic fields. In fact, we give several families of real biquadratic fields such that and , where and are the -class group and the -Iwasawa module of respectively. Moreover, Greenberg's conjecture is verified for some new families of number fields; in particular, we determine the complete list of all real biquadratic fields with trivial -Iwasawa module. This work is a continuation of M. M. Chems-Eddin, Greenberg's conjecture and Iwasawa module of real biquadratic fields I, J. Number Theory, 281 (2026), 224-266.
Paper Structure (8 sections, 23 theorems, 34 equations, 1 figure)

This paper contains 8 sections, 23 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['maintheorem']}
  3. Prerequisites
  4. The proof of Theorem \ref{['maintheorem']}
  5. Some infinite families of real biquadratic fields $K$ such that $\mathrm{rank}(A(K))=\mathrm{rank}(A_\infty(K))=3$
  6. Investigation of Iwasawa module and Greenberg's conjecture
  7. Triviality of Iwasawa module of real biquadratic fields
  8. Greenbergs conjecture over the fields $K_\nu=\mathbb{Q}(\sqrt{\nu q},\sqrt{rs})$, with $\nu\in\{1,2\}$

Key Result

Theorem 1.2

Let $K$ be a real biquadratic number field that is of the form $D)$ or $F)$ with $K\not=L$. Then $\mathrm{rank}(A(K_\infty))\leq 2$ and $\mathrm{rank}(A(K_\infty))=\mathrm{rank}( A(K))$ if and only if $K$ takes one of the following forms:

Figures (1)

  • Figure 1: Unramified extensions

Theorems & Definitions (38)

  • Theorem 1.2: The First Main Theorem
  • Lemma 2.1: Ku-50
  • Lemma 2.2: fukuda
  • Lemma 2.3: Qinred, Lemma 2.4
  • Proposition 2.4: BLS98, Proposition 7
  • Theorem 2.5: aaboune, Theorem 4.10
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 28 more