Stabilization for Iwasawa modules in ${\mathbb{Z}}_p$-extensions

Abstract

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 $k_m$ ($m⩾n$). We study the capitulation kernels $H_{n,m}:=\ker (i_{n,m})$ and $H_n:={\bigcup }_{m⩾n}{H}_{n,m}$ 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 $K/k$.