On $\mathbb{Z}_2$-extensions of real quadratic fields with class group of $2$-rank three

Abstract

In this paper, we study the unramified Iwasawa module over the cyclotomic $\mathbb{Z}_2$-extension of the 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. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The criterion is based on the nonexistence of a $2$-group with a certain prescribed quotient. Using this criterion, we construct an infinite family of such real quadratic fields with unramified Iwasawa module of type $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $2$-class group of type $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. This gives the first example of an infinite family of real quadratic fields whose ideal class group has $2$-rank three and whose cyclotomic $\mathbb{Z}_2$-extension is totally ramified and satisfies Greenberg's conjecture.