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March 2016 On the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{p})$, III
Takashi Fukuda, Keiichi Komatsu, Manabu Ozaki, Takae Tsuji
Funct. Approx. Comment. Math. 54(1): 7-17 (March 2016). DOI: 10.7169/facm/2016.54.1.1

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 $10^5$.

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Takashi Fukuda. Keiichi Komatsu. Manabu Ozaki. Takae Tsuji. "On the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{p})$, III." Funct. Approx. Comment. Math. 54 (1) 7 - 17, March 2016. https://doi.org/10.7169/facm/2016.54.1.1

Information

Published: March 2016
First available in Project Euclid: 22 March 2016

zbMATH: 06862330
MathSciNet: MR3477730
Digital Object Identifier: 10.7169/facm/2016.54.1.1

Subjects:
Primary: 11R23
Secondary: 11Y40

Keywords: cyclotomic unit , Iwasawa invariant , real quadratic field

Rights: Copyright © 2016 Adam Mickiewicz University

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Vol.54 • No. 1 • March 2016
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