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On the $\mathfrak{M}_H(G)$-property

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  24 June 2025

SÖREN KLEINE ,
AHMED MATAR  and
SUJATHA RAMDORAI
SÖREN KLEINE
Affiliation:
Institut für Anwendungssicherheit, Fakultät für Informatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85579 Neubiberg, Germany. e-mail: soeren.kleine@unibw.de
AHMED MATAR
Affiliation:
Department of Mathematics, University of Bahrain, P.O. Box 32038, Sukhair, Bahrain. e-mail: amatar@uob.edu.bh
SUJATHA RAMDORAI
Affiliation:
Department of Mathematics, 1984, Mathematics Road, University of British Columbia, Vancouver, V6T1Z2, Canada. e-mail: sujatha@math.ubc.ca
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Abstract

Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p. Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property for ${{\mathbb{Z}}}_p$-extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$-extension $K_\infty/K$. The equivalent conditions involve the growth of $\mu$-invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$-extensions in $K_\infty$, and the boundedness of $\lambda$-invariants as one runs over ${{\mathbb{Z}}}_p$-extensions of K inside of $K_\infty$.

Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$-extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$-property sometimes can be shifted to a larger base field K′.

MSC classification

Primary: 11R23: Iwasawa theory

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 53
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

Dedicated to the memory of John H. Coates

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