Greenberg's conjecture for real quadratic fields and the cyclotomic ${\mathbb{Z}}_2$-extensions

Let ${\mathcal{A}}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic ${\mathbb{Z}}_2$-extension of a real quadratic number field $F$. The cardinality of ${\mathcal{A}}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the ${\mathcal{A}}_n$’s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg’s conjecture holds for those fields.