On the Iwasawa invariants of non-cotorsion Selmer groups

We study the variation of Iwasawa invariants of Selmer groups and fine Selmer groups of abelian varieties over $\mathbb{Z}_p$-extensions of a fixed number field $K$. It is shown that the $\lambda$-invariants can be unbounded if the $\Lambda$-coranks of the Selmer groups (respectively fine Selmer groups) vary. In contrast, the classical Iwasawa $\lambda$-invariants of $\mathbb{Z}_p$-extensions are expected to be bounded, at least for small base fields like imaginary quadratic fields. For fine Selmer groups, the boundedness of $\lambda$-invariants is related to the (possible) failure of the weak Leopoldt conjecture.

Keywords

boundedness of Iwasawa invariants, abelian varieties, Selmer groups, fine Selmer groups, weak Leopoldt conjecture

2010 Mathematics Subject Classification

11R23

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Received 22 September 2021

Accepted 7 February 2022

Published 6 March 2023