Unramified Iwasawa module of ${\mathbb{Z}}_2$-extension of certain quadratic fields with a bounded quotient

We consider an infinite family of real quadratic fields $k$ where the discriminant has three distinct odd prime factors, and the prime 2 splits. We show that the unramified Iwasawa module $X(k_{\mathrm{\infty }})$ associated with the ${\mathbb{Z}}_2$-extension of $k$ has a bounded quotient. Thus, we also verify Greenberg’s conjecture on the vanishing of Iwasawa invariants for such fields and obtain a finer structure for $X(k_{\mathrm{\infty }})$.