We study the generalised Iwasawa invariants of -extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion -modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of -extensions of K, i.e., that the generalised Iwasawa invariants of a -extension of K bound the invariants of all -extensions of K in an open neighbourhood of . Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain -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.
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