Functiones et Approximatio Commentarii Mathematici

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

Takashi Fukuda, Keiichi Komatsu, Manabu Ozaki, and Takae Tsuji

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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 $10^5$.

Article information

Funct. Approx. Comment. Math., Volume 54, Number 1 (2016), 7-17.

First available in Project Euclid: 22 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11Y40: Algebraic number theory computations

Iwasawa invariant cyclotomic unit real quadratic field


Fukuda, Takashi; Komatsu, Keiichi; Ozaki, Manabu; Tsuji, Takae. On the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{p})$, III. Funct. Approx. Comment. Math. 54 (2016), no. 1, 7--17. doi:10.7169/facm/2016.54.1.1.

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