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.



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


