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 \(k\)).


11R23 Iwasawa theory
11R18 Cyclotomic extensions


Zbl 0334.12013


Full Text: DOI


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