## On tame pro-$$p$$ Galois groups over basic $$\mathbb Z_p$$-extensions.(English)Zbl 1291.11131

Let $$p$$ be a prime, $$k$$ a number field and $$S$$ a finite set of places of $$k$$. Denote by $$G_S(k)$$ the Galois group of the maximal pro-$$p$$-extension $$k_S / k$$ unramified outside $$S$$ and the infinite places. Let $$k_{\infty}$$ be the cyclotomic $$\mathbb Z_p$$-extension of $$k$$. If $$S$$ contains all $$p$$-adic places of $$k$$, then $$G_S(k)$$ and its subgroup $$G_S(k_{\infty})$$ have been studied extensively in Iwasawa theory. The paper under review focuses on the case, where $$k = \mathbb Q$$ and $$p \not\in S$$.
More precisely, let $$p$$ be an odd prime and let $$S = \left\{q_1, q_2 \right\}$$ be a set of two primes such that $$q_i \equiv 1 \mod p$$ (note that only primes $$q_i \equiv 1 \mod p$$ can (tamely) ramify in a $$p$$-extension of $$\mathbb Q$$). Suppose further that $$q_i \not\equiv 1 \mod p^2$$ and $$G_{\emptyset}(K_{\infty}) = 1$$, where $$K := \mathbb Q_{\left\{q_1\right\}} \mathbb Q_{\left\{q_2\right\}}$$. Then the main result of this article states that $$G_S(\mathbb Q_{\infty})$$ is a metacyclic pro-$$p$$-group: $G_S(\mathbb Q_{\infty}) = \langle a,b \mid a^{p^2} = 1, \quad b^{-1} a b = a^{1+p} \rangle.$ Moreover, when $$\gamma$$ is a topological generator of $$\Gamma := \mathrm{Gal}(\mathbb Q_{\infty} / \mathbb Q)$$, then $$\gamma$$ acts on $$G_S(\mathbb Q_{\infty})$$ as $$a^{\gamma} = a$$ and $$b^{\gamma} = b^{1+p} a^p$$. A similar result holds when $$p=2$$.
Finally, the authors show that the above hypotheses imply that $$G_{\emptyset}(k_{\infty})^{ab}$$ is finite for every finite subextension $$k / \mathbb Q$$ inside $$\mathbb Q_{\infty, S}$$ as conjectured by R. Greenberg [Am. J. Math. 98, 263–284 (1976; Zbl 0334.12013)] (for any totally real number field $$k$$).

### MSC:

 11R23 Iwasawa theory 11R18 Cyclotomic extensions

Zbl 0334.12013

PARI/GP
Full Text:

### References:

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