Local behaviour of generalised Iwasawa invariants

Abstract

We study the local behaviour of generalised Iwasawa invariants attached to multiple $$\mathbb {Z}_p$$ -extensions of a number field K, with respect to a suitable topology on the sets of $$\mathbb {Z}_p^d$$ -extensions of K, $$d \in \mathbb {N}$$ . These invariants generalise the classical Iwasawa invariants which describe the asymptotic growth of the ideal class groups of the intermediate fields in a $$\mathbb {Z}_p$$ -extension. Generalising work for $$\mathbb {Z}_p$$ -extensions and classical Iwasawa invariants, we show that, under certain assumptions, generalised Iwasawa invariants are locally maximal. In other words, the generalised Iwasawa invariants of certain $$\mathbb {Z}_p^d$$ -extensions $$\mathbb {K}$$ of K bound the generalised Iwasawa invariants for all $$\mathbb {Z}_p^d$$ -extensions of K close to $$\mathbb {K}$$ . The main ingredient in the proof is the precise investigation of the Galois module structure of the ideal class groups, which is much more involved than in the one-dimensional case.