Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields
H Laxmi, Anupam Saikia
TL;DR
This work investigates Greenberg's conjecture for the cyclotomic $\mathbb{Z}_2$-extension of the real biquadratic field $K=\mathbb{Q}(\sqrt{p},\sqrt{r})$ under a specified congruence condition. By combining genus theory, class-field theory, and capitulation results, the authors establish that the Iwasawa module $X(K_{\infty})$ is finite and cyclic with $\lambda(K_{\infty}/K)=0$, and they bound the growth of the $2$-class groups across the layers $K_n$ via $\#A(\mathbb{Q}_n(\sqrt{p})) \le \#A(K_n) \le 2\cdot\#A(\mathbb{Q}_n(\sqrt{p}))$. They derive an equivalent criterion for when $\#A(K_n)$ equals $\#A(\mathbb{Q}_n(\sqrt{p}))$, link this to capitulation phenomena, and analyze the order of $A(K_1)$ in a key subcase where $p = a^2 - 2b^2$ with $\left(\dfrac{a}{p}\right)=-1$, showing $\#A(K_1)=2$ and capitulation. The results contribute concrete structural insights into the $2$-class tower of this family and support Greenberg-type stability for these real multiquadratic fields, with implications for ray class groups and the behavior of Hilbert class fields in the $2$-adic setting.
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.