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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

Citations:

Zbl 0334.12013

Software:

PARI/GP
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References:

[1] Blackburn, N., On prime-power groups in which the derived group has two generators, Proc. Camb. Philos. Soc., 53, 19-27, (1957) · Zbl 0077.03202
[2] Boston, N., Galois groups of tamely ramified \(p\)-extensions, J. Théor. Nr. Bordx., 19, 59-70, (2007) · Zbl 1123.11038
[3] Fujii, S., On the maximal pro-\(p\) extension unramified outside \(p\) of an imaginary quadratic field, Osaka J. Math., 45, 41-60, (2008) · Zbl 1143.11041
[4] Gras G.: Class Field Theory—From Theory to Practice. Springer, Berlin (2003) · Zbl 1019.11032
[5] Greenberg, R., On the Iwasawa invariants of totally real number fields, Am. J. Math., 98, 263-284, (1976) · Zbl 0334.12013
[6] Hajir, F.: Tame pro-p Galois groups: A survey of recent work. In: AGCT 2003, séminaires et congrés 11, Société Mathématique de France, pp. 111-124 (2005) · Zbl 1155.11028
[7] Koch H.: Galois Theory of \(p\)-extensions. Springer, Berlin (2002) · Zbl 1023.11002
[8] Komatsu, K., On the maximal \(p\)-extensions of real quadratic fields unramified outside \(p\), J. Algebra, 123, 240-247, (1989) · Zbl 0682.12005
[9] Kubotera, N., Greenberg’s conjecture and leopoldt’s conjecture, Proc. Japan Acad. Ser. A Math. Sci., 76, 108-110, (2000) · Zbl 0971.11053
[10] Mizusawa, Y., On the maximal unramified pro-2-extension over the cyclotomic \({\mathbb Z_2}\)-extension of an imaginary quadratic field, J. Théor. Nr. Bordx., 22, 115-138, (2010) · Zbl 1221.11215
[11] Mouhib, A.; Movahhedi, A., On the \(p\)-class tower of a \({\mathbb Z_p}\)-extension, Tokyo J. Math., 31, 321-332, (2008) · Zbl 1209.11095
[12] Neukirch J., Schmidt A., Wingberg K.: Cohomology of number fields, second edition (Grundlehren der Mathematischen Wissenschaften 323). Springer, Berlin (2008) · Zbl 1136.11001
[13] Okano, K., Abelian \(p\)-class field towers over the cyclotomic \({\mathbb Z_p}\)-extensions of imaginary quadratic fields, Acta Arith., 125, 363-381, (2006) · Zbl 1155.11051
[14] Ozaki, M.; Taya, H., On the Iwasawa \(λ\)_{2}-invariants of certain families of real quadratic fields, Manuscr. Math., 94, 437-444, (1997) · Zbl 0935.11040
[15] Ozaki, M., Non-abelian Iwasawa theory of \({\mathbb Z_p}\)-extensions, J. Reine Angew. Math., 602, 59-94, (2007) · Zbl 1123.11034
[16] Salle, L., Sur LES pro-\(p\)-extensions à ramification restreinte au-dessus de la \({\mathbb Z_p}\)-extension cyclotomique d’un corps de nombres, J. Théor. Nr. Bordx., 20, 485-523, (2008) · Zbl 1163.11071
[17] Salle, L., On maximal tamely ramified pro-2-extensions over the cyclotomic \({\mathbb Z_2}\)-extension of an imaginary quadratic field, Osaka J. Math., 47, 921-942, (2010) · Zbl 1263.11097
[18] Sharifi, R.T., On Galois groups of unramified pro-\(p\) extensions, Math. Ann., 342, 297-308, (2008) · Zbl 1165.11078
[19] Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., 62, 181-234, (1980) · Zbl 0465.12001
[20] Taya, H., Yamamoto, G.: Notes on certain real abelian 2-extension fields with \(λ\)_{2} = \(μ\)_{2} = \(ν\)_{2} = 0. Trends in Mathematics. Information Center for Mathematical Sciences, vol. 9(1), pp. 81-89 (2006). http://mathnet.kaist.ac.kr/new_TM/ · Zbl 1221.11215
[21] The Pari Group, PARI/GP. Bordeaux (2008). http://pari.math.u-bordeaux.fr/ · Zbl 0682.12005
[22] Washington L.C.: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Math. vol. 83. Springer, Berlin (1997) · Zbl 0966.11047
[23] Yamamoto, G., On the vanishing of Iwasawa invariants of absolutely abelian \(p\)-extensions, Acta Arith., 94, 365-371, (2000) · Zbl 0964.11048
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