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Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. $E[p^\infty](F_\infty)$ is finite ?

Proofs or references are welcome.

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Yes. If $E$ has potentially good reduction, this is due to H. Imai, Proc Japan Adac Math Sci 51 (1975). A non-standard proof is Theorem A.2.8 in Coates-Sujatha's "Galois cohomology of elliptic curves"

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  • $\begingroup$ Is it true if $E$ has supersingular reduction at $p$ ? $\endgroup$
    – Robert
    Mar 17, 2016 at 20:09
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    $\begingroup$ Yes, they don't assume anything on $E$. It has more to do with the shape of the image of the global Galois representation on $T_pE$ rather than how this looks locally at $p$. In particular their proof splits up according to whether $E$ has cm or not. $\endgroup$ Mar 17, 2016 at 20:11

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