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Fine Selmer groups and ideal class groups

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Abstract

Let K be a number field, let A be an abelian variety defined over K and let \(K_\infty /K\) be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If \(K_\infty \) contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa \(\mu \)-invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple \(\mathbb {Z}_p\)-extensions), we can also prove relations between suitably generalised Iwasawa \(\lambda \)-invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa \(\lambda \)-invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple \(\mathbb {Z}_p\)-extensions.

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Notes

  1. Actually Perbet considered, more generally, pro-p-groups which are p-valued. This includes the uniform p-groups (see [10,  p. 81]).

  2. This lemma is stated only for finite groups, but it can be generalised to pro-p groups easily.

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Acknowledgements

We thank Meng Fai Lim and Debanjana Kundu for kindly sending us a preprint of their work and answering our questions. We would also like to thank Damaris Schindler for supporting a visit of the first author in Göttingen. Part of this research was conducted while the second author was a postdoc at the University of Göttingen. Finally, we are grateful to the anonymous referee for the very quick and thorough reading of our lengthy manuscript.

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Authors and Affiliations

  1. Institut für Theoretische Informatik, Mathematik und Operations Research, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    Sören Kleine

  2. Department of Mathematics and Statistics, Université Laval, Québec City, Canada

    Katharina Müller

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  1. Sören Kleine
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  2. Katharina Müller
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Correspondence to Katharina Müller.

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The second named author was supported by Antonio Lei’s NSERC Discovery Grants RGPIN-2020-04259 and RGPAS-2020-00096.

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Kleine, S., Müller, K. Fine Selmer groups and ideal class groups. Ramanujan J 61, 1213–1267 (2023). https://doi.org/10.1007/s11139-022-00619-8

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