Rendiconti del Seminario Matematico della Università di Padova

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Volume 136, 2016, pp. 137–155
DOI: 10.4171/RSMUP/136-10

Published online: 2016-12-22

Stabilization for Iwasawa modules in $\mathbb Z_p$-extensions

Andrea Bandini[1] and Fabio Caldarola[2]

(1) Università degli Studi di Parma, Italy
(2) Università degli Studi della Calabria, Arcavacata di Rende (Cosenza), Italy

Let $K/k$ be a $\mathbb Z_p$-extension of a number field $k$ with layers $k_n$. Let $i_{n,m}$ be the map induced by inclusion between the $p$-parts of the class groups of $k_n$ and $k_m$ ($m \geqslant n$). We study the capitulation kernels $H_{n,m}:=\mathrm {ker} (i_{n,m})$ and $H_n:=\bigcup_{m \geqslant n}H_{n,m}$ to give some explicit formulas for their size and prove stabilization properties for their orders and $p$-ranks. We also briefly investigate stabilization properties for the cokernel of $i_{m,n}$ and for the kernels of the norm maps and point out their relations with the nullity of the Iwasawa invariants for $K/k$.

Keywords: $\mathbb Z_p-extensions, Iwasawa modules, Iwasawa invariants

Bandini Andrea, Caldarola Fabio: Stabilization for Iwasawa modules in $\mathbb Z_p$-extensions. Rend. Sem. Mat. Univ. Padova 136 (2016), 137-155. doi: 10.4171/RSMUP/136-10