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