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

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 $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(A(k_{\mathrm{\infty }}))=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(A(k_1))$?, where $A(k_{\mathrm{\infty }})$ 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 $A(k_{\mathrm{\infty }})$ is trivial or isomorphic to $\mathbb{Z}\mathrm{/}{2}^n\mathbb{Z}$ or $\mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}{2}^n\mathbb{Z}$, 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.