Fujii, Satoshi Non-abelian Iwasawa theory of cyclotomic \(\mathbb Z_p\)-extensions. (English) Zbl 1159.11041 The COE seminar on mathematical sciences 2007. Papers from the seminar held in Keio, Japan, 2007. Yokohama: Keio University, Department of Mathematics. Seminar on Mathematical Sciences 37, 85-97 (2008). Let \(k_\infty/k\) be the cyclotomic \({\mathbb Z}_p\)-extension of a number field \(k.\) This paper is a survey of some recent results on the structure of the Galois group \(\widetilde G_\infty\) of the maximal unramified pro-\(p\)-extension of \(k_\infty.\) One main body of information concerns the description of \(G_\infty\) by generators and relations when \(k\) is an imaginary quadratic field \({\mathbb Q}(\sqrt{-m})\): all cases for which \(\widetilde G_\infty\) is abelian are known (Mizusawa, Ozaki, Okano); when \(\widetilde G_\infty\) is not abelian, relations are known in special cases (for \(p = 2,\) for particular families of \(m\) and small values of the \(\lambda\)-inviariant of \(\widetilde G_\infty^{ab}).\) A second body of results concerns the pro-\(p\)-freeness of \(\widetilde G_\infty.\) The author shows two cases when \(\widetilde G_\infty\) is non-abelian and non-free: – when \(p\) splits totally in \(k\) and the generalized Greenberg’s conjecture holds, – when \(k\) is an imaginary quadratic field which is decomposed at \(p\) and which admits a \({\mathbb Z}_p\)-extension \(K\) such that all \(p\)-primes of \(K\) are ramified in \(K/k\) and the unramified Iwasawa module of \(K/k\) contains a non-trivial finite submodule.For the entire collection see [Zbl 1144.00006]. Reviewer: Thong Nguyen Quang Do (Besançon) Cited in 1 ReviewCited in 1 Document MSC: 11R23 Iwasawa theory 11R18 Cyclotomic extensions Keywords:\(p\)-class field tower PDFBibTeX XMLCite \textit{S. Fujii}, Semin. Math. Sci. 2007, 85--97 (2008; Zbl 1159.11041)