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

In this paper we are interested in the stability of the $2$-rank of the class group in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic fields. In fact, we give several families of real biquadratic fields $K$ such that $rank(A(K)) =rank(A_\infty(K))$ and $rank(A(K))\leq 3$, where $A(K)$ and $A_\infty(K)$ are the $2$-class group and the $2$-Iwasawa module of $K$ respectively. Moreover, Greenberg's conjecture is verified for some new families of number fields; in particular, we determine the complete list of all real biquadratic fields with trivial $2$-Iwasawa module. This work is a continuation of M. M. Chems-Eddin, Greenberg's conjecture and Iwasawa module of real biquadratic fields I, J. Number Theory, 281 (2026), 224-266.