Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields

TL;DR

This paper investigates the stability of 2-class groups in the cyclotomic $Z_2$-extension of certain real biquadratic fields, providing evidence for Greenberg's conjecture through class field theory and group theoretic methods.

Contribution

It extends the verification of Greenberg's conjecture to an infinite family of real biquadratic fields using elementary group and class field theoretic techniques.

Findings

Confirmed stability of 2-class groups in the studied family

Linked capitulation phenomena to class group sizes

Supported Greenberg's conjecture for these fields

Abstract

Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we consider an infinite family of real biquadratic fields $K$. With some extensive use of elementary group theoretic and class field theoretic arguments, we investigate the $2$-class groups of the $n$-th layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of $K$ and verify Greenberg's conjecture. We also relate capitulation of ideal classes of certain sub-extensions of $K_n$ to the relative sizes of the $2$-class groups.