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Bounding the Iwasawa invariants of Selmer groups

Part of: Theory of modules and ideals Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  29 June 2020

Sören Kleine
Sören Kleine*
Affiliation:
Institut für Theoretische Informatik, Mathematik und Operations Research, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany
*
e-mail: soeren.kleine@unibw.de
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Abstract

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

Keywords

Fukuda moduleSelmer groupMordell–Weil rankfine Selmer group

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$ 13C12: Torsion modules and ideals
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1390 - 1422
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Copyright
© Canadian Mathematical Society 2020

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