Stability of 2-class groups in $\mathbb{Z}_2$-extension of certain real quadratic number fields

H Laxmi, Anupam Saikia

Published 2023-10-05Version 1

For a real quadratic field $K= \mathbb{Q}(\sqrt{d})$ with $d$ having three distinct prime factors, it has been proven that under certain conditions, the Iwasawa module $X_{\infty}$ corresponding to the cyclotomic $\mathbb{Z}_2$-extension of $K$ is cyclic. In this paper, with some elementary assumptions on the prime factors of $d$, we show that $X_{\infty}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. Consequently, we prove that the Iwasawa $\lambda$-invariant for such fields is equal to 0, validating Greenberg's conjecture for these fields.