## 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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