On real biquadratic number fields for which the $2$-Iwasawa module has rank $3$

Source abstract

We investigate the stability of the $2$-rank in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic number fields. We determine several families of real biquadratic fields $F$ for which the ranks of the $2$-class group $A(F)$ and the associated $2$-Iwasawa module $A_\infty(F)$ are both equal to $3$. More precisely, we provide a complete classification of all fields $F=\mathbb{Q}(\sqrt{q},\sqrt{d}),$ where $q\equiv 3\pmod{4}$ is a prime number and $d$ is an odd positive square-free integer, and all fields $F=\mathbb{Q}(\sqrt{2q},\sqrt{d}),$ where $q\equiv 3\pmod{4}$ is a prime number and $d\equiv 1\pmod{4}$ is a positive square-free integer, for which $\operatorname{rank}(A(F))=\operatorname{rank}(A_\infty(F))=3.$