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

Abstract: The main aim of this paper is to investigate the Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem: $$\text{What are real biquadratic number fields } K\text{ such that } rank(A(k_\infty))=rank(A(k_1))? $$ where $A(k_\infty)$ are 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 $\mathbb Z/2\mathbb Z^n \times\mathbb Z/2^{m} \mathbb Z$, where $n$ and $m$ are given positive integers. The reader can find also some results concerning $2$-rank of the class group of certain real triquadratic fields.