On restricted ramifications and pseudo-null submodules of Iwasawa modules for \(\mathbb{Z}^2_p\)-extensions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1316.11098
author: Satoshi Fujii
publication date: 2014-11-04T00:00:00Z
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 6365670
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Greenberg's conjecture
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: \(\mathbb Z^2_p\)-extensions
zbMATH Keywords: Iwasawa modules
MaRDI profile type: Publication
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title: On restricted ramifications and pseudo-null submodules of Iwasawa modules for \(\mathbb{Z}^2_p\)-extensions
published in: Journal of the Ramanujan Mathematical Society
review text: Let \(p\) be an odd prime, \(k/{\mathbb Q}\) a finite extension and \(k_{\infty}/k\) the cyclotomic \({\mathbb Z}_p\)-extension of \(k\). Let \(L(k_{\infty})/k_{\infty}\) be the maximal unramified pro-\(p\)-abelian extension. The complete group ring \({\mathbb Z}_p[[\roman{Gal}( k_{\infty}/k)]]\) acts naturally on \(\roman{Gal}(L(k_{\infty})/k_{ \infty})\). \textit{M. Ozaki} proved in [Tohoku Math. J. (2) 49, No. 3, 431--435 (1997; Zbl 0896.11044)] that if \(k\) is totally real and \(p\) splits completely in \(k/{\mathbb Q}\), then \(\roman{Gal}(L(k_{\infty})/ k_{\infty})\neq 0\) if and only if \(p\zeta_p(0,k)\equiv 0\bmod p\), where \(\zeta_p(s,k)\) denotes the \(p\)-adic zeta function of \(k\). As a consequence, Ozaki proved that if \(k\) is totally real, \(p\) splits completely in \(k/{\mathbb Q}\) and Leopoldt's conjecture holds for \(p\) and \(k\), then \(\roman{Gal}(L(k_{\infty})/k_{\infty})\) has a non-trivial finite \({\mathbb Z}_p[[\roman{Gal}(k_{\infty}/k)]]\)-submodule if and only if \(M(k_{\infty})\neq L(k_{\infty})\). Here \(M(k_{\infty})\) denotes the maximal abelian pro-\(p\)-extension of \(k_{\infty}\) unramified outside \(p\). It is a conjecture of \textit{R. Greenberg} [Am. J. Math. 98, 263--284 (1976; Zbl 0334.12013)] that for totally real fields \(k\), \(\text{Gal}(L(k_{\infty})/k_{\infty})\) is finite.NEWLINENEWLINEIn this paper, the author shows results analogous to the ones obtained by Ozaki for \({\mathbb Z}_p^2\)-extensions of imaginary quadratic fields. Namely, consider \(k\) an imaginary quadratic field such that \(p\) splits. Let \(\tilde{k}/k\) be the unique \({\mathbb Z}_p^2\)-extension. Let \(L(\tilde{k})/\tilde{k}\) be the maximal unramified pro-\(p\)-abelian extension. Let \(\chi\) be the Dirichlet character associated to \(k\) and \(\omega\) the Teichmüller character of \({}\bmod p\). Let \(L_p(s,\omega\chi^{-1})\) be the \(p\)-adic Dirichlet \(L\)-function associated to \(\omega \chi^{-1}\). The author shows that \(\roman{Gal}( L(\…
reviewed by: Gabriel D. Villa Salvador
description: scientific article; zbMATH DE number 6365670