On restricted ramifications and pseudo-null submodules of Iwasawa modules for \(\mathbb{Z}^2_p\)-extensions (Q2927747)

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 \(\roman{Gal}( L(\tilde{k})/\tilde{k})\neq 0\) if and only if \(\frac{L^{\prime}_p(0, \omega\chi^{-1})}{p}\equiv 0 \bmod p\).NEWLINENEWLINEThe second main result of this paper is that \(\roman{Gal}(L( \tilde{k})/\tilde{k})\) has a non-trivial pseudo-null \({\mathbb Z}_p[[ \roman{Gal}(\tilde{k}/k)]]\)-submodule if and only if \(M_{\mathfrak p} (\tilde{k})\neq L(\tilde{k})\), where \(M_{\mathfrak p}(\tilde{k})/\tilde{k}\) denotes the maximal pro-\(p\)-abelian extension unramified outside all primes lying above \({\mathfrak p}\) and \((p)={\mathfrak p} {\mathfrak p}^{\prime}\) in \(\tilde{k}\). A finitely generated, torsion \({\mathbb Z}_p[[\roman{Gal}(\tilde{k}/k)]]\)-module is called pseudo-null if the annihilator ideal contains two relatively prime elements.NEWLINENEWLINEOne of the main tools in the proof of the second main result is that \(\roman{Gal}(M_{\mathfrak p}(\tilde{k})/\tilde{k})\) has no non-trivial pseudo-null \({\mathbb Z}_p[[\roman{Gal}(\tilde{k}/k)]]\)-submodules. This result was obtained by \textit{B. Perrin-Riou} [Mém. Soc. Math. Fr., Nouv. Sér. 17, 130 p. (1984; Zbl 0599.14020)] but a proof is presented here.