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Fine Selmer groups of congruent p-adic Galois representations

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  29 September 2021

Sören Kleine Open the ORCID record for Sören Kleine [Opens in a new window]  and
Katharina Müller Open the ORCID record for Katharina Müller [Opens in a new window]
Sören Kleine*
Affiliation:
Institut für Theoretische Informatik, Mathematik und Operations Research, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, Neubiberg85577, Germany
Katharina Müller
Affiliation:
Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, Göttingen 37073, Germany e-mail: katharina.mueller@mathematik.uni-goettingen.de
*
e-mail: soeren.kleine@unibw.de
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Abstract

We compare the Pontryagin duals of fine Selmer groups of two congruent p-adic Galois representations over admissible pro-p, p-adic Lie extensions $K_\infty $ of number fields K. We prove that in several natural settings the $\pi $ -primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $\mu $ -invariants. In the special case of a $\mathbb {Z}_p$ -extension $K_\infty /K$ , we also compare the Iwasawa $\lambda $ -invariants of the fine Selmer groups, even in situations where the $\mu $ -invariants are nonzero. Finally, we prove similar results for certain abelian non-p-extensions.

Keywords

Admissible p-adic Lie extensionabelian varietyp-adic Galois representationfine Selmer groupIwasawa invariants

MSC classification

Primary: 11R23: Iwasawa theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 702 - 722
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Copyright
© Canadian Mathematical Society 2021

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