On $$\mathbb {Z}_2$$-extensions of real quadratic fields with four odd ramified prime numbers

In this paper, we study the Iwasawa $$\lambda $$ -invariant of the cyclotomic $$\mathbb Z _2$$ -extension of a real quadratic field $$\mathbb Q (\sqrt{p_1p_2p_3p_4})$$ , where $$p_1, p_2, p_3$$ and $$p_4$$ are distinct odd prime numbers satisfying certain arithmetic conditions. We give a new infinite family of real quadratic fields for which Greenberg’s conjecture holds.