On the maximal finite Iwasawa submodule in Zp-extensions and capitulation of ideals

For Zp-extensions of a number fields the properties of stabilization and capitulation of ideal classes are of great interest and are also related to very important aspects and problems such as Greenberg’s conjecture. In [3] these properties are deeply investigated from the point of view of the maximum finite submodule of the Iwasawa module and new invariants and parameters are introduced to give precise characterizations of these phenomena. In this article we will discuss some bounds that control the increment of the index that measures the capitulation delay in the tower and moreover we will prove how some results on the capitulation kernels in [3] have to be considered optimal. Finally, we will also give some further applications and examples that emphasize the cases of false (or failed) stabilization in this context.