Generalised Iwasawa invariants and the growth of class numbers

  • 1 Institut für Theoretische Informatik, Mathematik und Operations Research, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany
Sören KleineORCID iD: https://orcid.org/0000-0002-6794-8061

Abstract

We study the generalised Iwasawa invariants of pd-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion p[[T1,,Td]]-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of pd-extensions of K, i.e., that the generalised Iwasawa invariants of a pd-extension 𝕂 of K bound the invariants of all pd-extensions of K in an open neighbourhood of 𝕂. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain p2-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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