On the $\Lambda$-cotorsion subgroup of the Selmer group

Let $E$ be an elliptic curve defined over a number field $K$ with supersingular reduction at all primes of $K$ above $p$. If $K_\infty / K$ is a $\mathbb{Z}_p$-extension such that $E(K_\infty) [p^\infty]$ is finite and $H^2 (G_S (K_\infty), E [p^\infty]) = 0$, then we prove that the $\Lambda$-torsion subgroup of the Pontryagin dual of $\operatorname{Sel}_{p^\infty} (E / K_\infty)$ is pseudo-isomorphic to the Pontryagin dual of the fine Selmer group of $E$ over $K_\infty$. This is the Galois-cohomological analog of a flat-cohomological result of Wingberg.

Keywords

elliptic curve, Selmer group, Iwasawa theory

2010 Mathematics Subject Classification

11G05, 11R23, 12G05

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Received 19 November 2018

Accepted 30 July 2019

Published 9 October 2020