## 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 $$10^5$$.

### MSC:

 11R23 Iwasawa theory 11Y40 Algebraic number theory computations

### Keywords:

Iwasawa invariant; cyclotomic unit; real quadratic field

### Citations:

Zbl 1215.11106; Zbl 1253.11098; Zbl 1358.11122
Full Text:

### References:

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