On the Iwasawa $\lambda$-invariant of the cyclotomic ${\mathbb{Z}}_2$-extension of a family of real quadratic fields in which $2$ splits

We study Greenberg’s conjecture for cyclotomic ${\mathbb{Z}}_2$-extensions of real quadratic fields. Let $K=\mathbb{Q}(\sqrt{pq})$, where $$p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}8,q\equiv 9\mathrm{m}\mathrm{o}\mathrm{d}16,\left(\frac{p}{q}\right)=-1.$$ Under the additional assumptions $${\left(\frac{2}{p}\right)}_4{\left(\frac{2}{q}\right)}_4{\left(\frac{pq}{2}\right)}_4=-1$$ and $${\left(\frac{2}{p}\right)}_4=-1\text{or}{\left(\frac{2}{q}\right)}_4=-1,$$ we prove that ${\lambda }_2(K)=0$. The proof combines Greenberg’s criterion for the split prime case with a capitulation argument modeled on Kumakawa. The main new input is a square-class computation of the Hasse unit index of the biquadratic extension $K_2=\mathbb{Q}(\sqrt{pq},\sqrt{2+\sqrt{2}})\mathrm{/}{\mathbf{Q}}_1=\mathbb{Q}(\sqrt{2})$, showing that $q(K_2)\le 2$.