Stabilization for Iwasawa modules in \(\mathbb{Z}_p\)-extensions (Q509922)

Let \(k_{\infty}/k\) be a \(\mathbb Z_p\)-extension of a number field \(k\) with \(n\)-th layer \(k_n\). Let \(A_n\) be the \(p\)-part of the class group of \(k_n\). By a famous theorem of Iwasawa one has \[ |A_n| = p^{e_n}, \, e_n = \lambda n + \mu p^n + \nu \] for sufficiently large \(n\), where \(\nu\) is a constant and \(\lambda\) and \(\mu\) are the Iwasawa invariants associated to the Iwasawa module \(X = \varprojlim_n A_n\). Suppose in the following (for simplicity) that all ramified primes in \(K/k\) are totally ramified. Then \textit{T. Fukuda} [Proc. Japan Acad. Ser. A Math. Sci. 70, 264--266 (1994; Zbl 0823.11064)] has shown that \(|A_n| = |A_{n+1}|\) for some \(n\) implies \(|A_n| = |A_m| = |X|\) for all \(m \geq n\). In the paper under review, the authors prove a similar result for capitulation kernels. For integers \(m \geq n\) let \(H_{n,m}\) be the kernel of the natural map \(A_n \rightarrow A_m\) induced by inclusion, and put \(H_n := \bigcup_{m \geq n} H_{n,m}\). Then \(|H_n| = |H_{n+1}|\) implies that \(H_n \simeq H_m \simeq D\) for all \(m \geq n\), where \(D\) denotes the maximal finite submodule of \(X\). In particular, there is an integer \(r\) such that \[ \dots < |H_{r-2}| < |H_{r-1}| < |H_r| = |H_{r+1}| = \dots = |D|. \] For any \(n \geq 0\) the authors then define an integer \(h(n)\). Here we only say that for \(n \geq r\) one has \(h(n) = n + \varepsilon\), where \(p^{\varepsilon}\) is the exponent of \(D\). Then for \(n < r\) one has \[ 1 = |H_{n,n}| \leq |H_{n,n+1}| \leq \dots \leq |H_{n,r}| < |H_{n,r+1}| < \dots < |H_{n,h(n)}| = |H_{n,h(n)+1}| = \dots \] whereas for \(n \geq r\) one has \[ 1 < |H_{n,n+1}| < \dots < |H_{n,h(n)}| = |H_{n,h(n)+1}| = \dots = |D|. \] In particular, \((|H_{n,m}|)_{m \geq n}\) stabilizes for \(m \geq h(n)\). In a final chapter, the authors deal with the relation between stabilization and the triviality of the Iwasawa invariants \(\lambda\) and \(\mu\) above. Similar results arising from the non-abelian theory of \textit{M. Ozaki} [J. Reine Angew. Math. 602, 59--94 (2007; Zbl 1123.11034)] are discussed in [the authors, Acta Arith. 169, No. 4, 319--329 (2015; Zbl 1331.11098)].