# Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

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.

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"
• Is it true if $E$ has supersingular reduction at $p$ ? Mar 17, 2016 at 20:09
• 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. Mar 17, 2016 at 20:11