On the Iwasawa \(\lambda\)-invariant of the cyclotomic \(\mathbb{Z}_2\)-extension of \(\mathbb{Q}(\sqrt{p})\). III. (English) Zbl 1407.11125

Summary: 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 Part I [the first two authors, Math. Comput. 78, No. 267, 1797–1808 (2009; Zbl 1215.11106)], [the first author, Interdiscip. Inf. Sci. 16, No. 1, 21–32 (2010; Zbl 1253.11098)] and Part II [the first two authors, Funct. Approximatio, Comment. Math. 51, No. 1, 167–179 (2014; Zbl 1358.11122)] 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 \(10^5\).


11R23 Iwasawa theory
11Y40 Algebraic number theory computations
Full Text: DOI Euclid


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