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A generalized Iwasawa’s theorem and its application. (English) Zbl 1472.11283

Let \(K/k\) be a \(\mathbb{Z}_p^d\)-extension of a number field \(k\) (\(p\in \mathbb{N}\) a prime) with Galois group \(\Gamma\) and associated Iwasawa algebra \(\Lambda:=\mathbb{Z}_p[[\Gamma]]\). For any intermediate number field \(k\subseteq E\subsetneq K\), let \(A_E\) be the \(p\)-part of the class group of \(E\) and \(A'_E\) the quotient of \(A_E\) modulo the classes generated by primes lying above \(p\). Consider the direct (resp. inverse) limits \(A_K\) and \(A'_K\) (resp. \(X_K\) and \(X'_K\)) of the \(A_E\) and \(A'_E\) with respect to the natural inclusions (resp. norm) maps. Those are modules over the Iwasawa algebra \(\Lambda\) and their structure is the main topic of various conjectures: most notably the Greenberg Generalized Conjecture, which predicts the pseudo-nullity of \(X_{\tilde{k}}\), with \(\tilde{k}\) the compositum of all the \(\mathbb{Z}_p\)-extensions of \(k\) (we recall that a finitely generated \(\Lambda\)-module \(M\) is pseudo-null, i.e., \(M\sim_\Lambda 0\), if it has at least two relatively prime annihilators).
The paper mainly deals with the subquotient of \(X_K\) (resp. \(X'_K\)) given by the inverse limit of the kernels of the capitulation maps \(c_{K/E}: A_E \longrightarrow A_K\) (resp. \(c'_{K/E}: A'_E \longrightarrow A'_K\)). Let \[ \dot{X}_K := \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}} \mathrm{Ker}\,c_{K/E}\quad \text{and}\quad \dot{X}'_K := \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}}\mathrm{Ker}\,c'_{K/E} \,,\] the authors prove that \(\dot{X}\sim_\Lambda 0\) and \(\dot{X}'\sim_\Lambda 0\), thus generalizing a result of I. Iwasawa [Ann. Math. (2) 98, 246–326 (1973: Zbl 0285.12008)] which was proved only for \(d=1\). The main technical point is a careful description of the topology of \(\Gamma\) (hence of \(\Lambda\)-modules) which is carried out by providing explicit generators for augmentation ideals of the group rings associated to the finite layers of \(K/k\). Once this is done, with the crucial ingredient of the \(\mathbb{Z}_p\)-flat sets described by P. Monsky [Math. Ann. 255, 217–227 (1981: Zbl 0437.12016)], the proof closely follows the classical one by Iwasawa.
In the final section the authors use the main theorem to obtain duality pseudo-isomorphisms \[ X_K^\# \sim_\Lambda \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}} A_E^\vee \quad\text{and}\quad (X'_K)^\# \sim_\Lambda \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}} (A'_E)^\vee, \] (where \(\,^\vee\) denotes the Pontrjagin dual and \(\,^\#\) the \(\Lambda\)-module with the twisted action induced by \(\gamma\longrightarrow \gamma^{-1}\)) via the theory of \(\Gamma\)-systems presented in [K.F. Lan et al., Trans. Am. Math. Soc. 370, No. 3, 1925–1958 (2018: Zbl 1444.11236)].

MSC:

11R29 Class numbers, class groups, discriminants
11R23 Iwasawa theory
12G05 Galois cohomology
11R65 Class groups and Picard groups of orders
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