On ${\mathbb{Z}}_2$-extensions of real quadratic fields

Let $k$ be a real quadratic field. Let $k_n$ be the $n$-th layer of the ${\mathbb{Z}}_2$-extension of $k$, and $2^{a_n}$ be the $2$-part of the class number of $k_n$. In this paper, we derive many laws concerning the sequence $(a_0,a_1,\dots )$, by iteratively extending the group of cyclotomic units.