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

2025/03/19 by Chems-Eddin, Mohamed Mahmoud · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.15727

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_∞)) = \rm rank(A(k₁))?, where A(k_∞) is the 2-Iwasawa module of k and A(k₁) is the 2-class group of k₁ the first layer of the cyclotomic \mathbb Z₂-extension of k. Moreover, we give several families of real biquadratic fields k such that A(k_∞) is trivial or isomorphic to \mathbb Z/2ⁿ \mathbb Z or \mathbb Z/2\mathbb Z ×\mathbb Z/2ⁿ \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.

Cited by

• Greenberg's conjecture and Inverse Galois problem: metacyclic-nonmodular group of type 1 whose abelianization is ℤ/2 ℤ×ℤ/2^m ℤ, m≥ 2

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• Iwasawa module of the cyclotomic ℤ₂-extension of certain real quadratic fields
• On the maximal unramified pro-2-extension of ℤ₂-extension of certain real biquadratic fields
• Stability of 2-class groups in the ℤ₂-extension of certain real biquadratic fields
• Unramified Iwasawa module of ℤ₂-extension of certain quadratic fields with a bounded quotient
• Greenberg's conjecture for real quadratic fields and the cyclotomic ℤ₂-extensions