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On the depth of the relations of the maximal unramified pro-\(p\) Galois group over the cyclotomic \(\mathbb Z_p\)-extension. (English) Zbl 1250.11094

Let \(p\) be a prime number, \(G\) a pro-\(p\) group, and \(1 \rightarrow R \rightarrow F \rightarrow G \rightarrow 1\) a presentation of \(G\) by a free pro-\(p\) group \(F\). Such a presentation is called minimal if inflation induces an isomorphism \(H^1(G,\mathbb Z/p) \rightarrow H^1(F,\mathbb Z/p)\). If the presentation is minimal, and if there is a non-negative integer \(i\) such that \(R \subseteq D_i(F)\) and \(R \subsetneq D_{i+1}(F)\), then \(i\) is called the derived depth of \(G\). The trivial group is said to have derived depth \(- \infty\), and the free pro-\(p\) group derived depth \(+ \infty\).
The author’s main result is the following. Let \(k\) be a number field in which \(p\) splits completely. Let \(k_\infty\) denote the cyclotomic \(\mathbb Z_p\)-extension of \(k\), let \(L/k_\infty\) be the maximal unramified pro-\(p\)-extension of \(k_\infty\), and let \(G(k_\infty)\) denote the Galois group of this extension. If Greenberg’s generalized conjecture holds for \(p\) and \(k\), then the derived depth of \(G(k_\infty)\) is at most \(1\), except when \(G(k_\infty) \simeq \mathbb Z_p\). At the end of the article, numerical examples involving complex quadratic base fields are given.

MSC:

11R23 Iwasawa theory
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