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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].

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions
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