Stabilization for iwasawa modules in ℤp-extensions

Let K/k be a ℤp-extension of a number field k with layers kn. Let in,m be the map induced by inclusion between the p-parts of the class groups of kn and km(m ≥ n). We study the capitulation kernels Hn,mWD ker(in,m) and Hn:= ∪m ≥ nHn;m to give some explicit formulas for their size and prove stabilization properties for their orders andp-ranks. We also briefly investigate stabilization properties for the cokernel of im;n and for the kernels of the norm maps and point out their relations with the nullity of the Iwasawa invariants for K/k.