On the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{p})$, III

Abstract

In the preceding papers, two of authors developed criteria for Greenberg conjecture of the cyclotomic $\mathbb{Z}_2$-extension of $k=\mathbb{Q}(\sqrt{p})$ with prime number $p$. Criteria and numerical algorithm in [5], [3] and [6] enable us to show $\lambda_2(k)=0$ for all $p$ less than $10^5$ except $p=13841, 67073$. All the known criteria at present can not handle $p=13841, 67073$. In this paper, we develop another criterion for $\lambda_2(k)=0$ using cyclotomic units and Iwasawa polynomials, which is considered a slight modification of the method of Ichimura and Sumida. Our new criterion fits the numerical examination and quickly shows that $\lambda_2(\mathbb{Q}(\sqrt{p}))=0$ for $p=13841, 67073$. So we announce here that $\lambda_2(\mathbb{Q}(\sqrt{p}))=0$ for all prime numbers $p$ less that $10^5$.