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

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 $\mathrm{rank}(A(k_{\infty }))=\mathrm{rank}(A(k_1))$?, 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.

42 pages. To appear in Journal of Number Theory