Generalised Iwasawa invariants and the growth of class numbers

Abstract

Abstract We study the generalised Iwasawa invariants of ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion ℤ p ⁢ [ [ T 1 , … , T d ] ] {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]} -modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of K, i.e., that the generalised Iwasawa invariants of a ℤ p d {\mathbb{Z}_{p}^{d}} -extension 𝕂 {\mathbb{K}} of K bound the invariants of all ℤ p d {\mathbb{Z}_{p}^{d}} -extensions of K in an open neighbourhood of 𝕂 {\mathbb{K}} . Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain ℤ p 2 {\mathbb{Z}_{p}^{2}} -extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.