A new approach to the investigation of Iwasawa invariants

Abstract

Let K be a fixed number field, let p be a prime number, and let Z_p denote the additive group of p-adic integers. The growth of the p-Sylow subgroups of the ideal class groups of the intermediate fields in a Z_p-extension L of K may be explicitly described in terms of the so-called Iwasawa invariants of L/K. In this thesis, we define a certain topology on the set of Z_p-extensions of the number field K, and we prove that Iwasawa's invariants are locally maximal with respect to this topology. Furthermore, we develop necessary and sufficient criteria for the invariants to be globally bounded. Finally, we generalise to the study of multiple Z_p-extensions and obtain similar results for generalised Iwasawa invariants. Our main tool is a method exploiting the stabilisation of certain ranks. This is a generalisation of a theorem of T. Fukuda.