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Greenberg’s generalized conjecture and unramified Galois groups over the cyclotomic \(\mathbb Z_p\)-extensions. (English) Zbl 1301.11073

From the text: Let \(p\) be a fixed prime number and \(k\) a number field. Recently, the studies of the Galois group \(G(k_\infty)\) of the maximal unramified pro-\(p\) extension \(\mathcal L(k_\infty)/k_\infty\) over the cyclotomic \(\mathbb Z_p\)-extension \(k_\infty\) of \(k\) are being developed. In this subject, to find number fields \(k\) such that \(G(k_\infty)\) is a non-abelian free pro-\(p\) group is a very important problem. However, it seems that there is no concrete such example of a number field \(k\). In this article, we show roughly if Greenberg’s generalized conjecture holds for \(p\) and \(k\) then \(G(k_\infty)\) can not be a non-abelian free pro-\(p\) group. We also show some examples of imaginary abelian fields \(k\).
This article is written as a report of the talk of the author at “Algebraic Number Theory and Related Topics”. Around the subjects of this article are already published [Interdiscip. Inf. Sci. 16, No. 1, 55–66 (2010; Zbl 1234.11145)] and [Acta Arith. 149, No. 2, 101-110 (2011; Zbl 1250.11094)] and published as a report [The COE seminar on mathematical sciences 2007. Sem. Math. Sci. 37, 85–97 (2008; Zbl 1159.11041)], so here we give a survey of the topics about the talk specifically.

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions
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