Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F\_n=F[E\_{p^n}]$ and $F\_{\infty}=F[E\_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F\_\infty/F)$ is an open subgroup of $GL\_2(\mathbb{Z}\_p)$. 


Consider the natural norm maps on $E\_{p^\infty}(F\_m)$. I read somewhere that for $m$ sufficiently large, these maps are essentially just multiplication by $p^4$ and so the projective limit of the inverse system $E\_{p^\infty}(F\_m)$ is zero. 


I don't know the proof of this. Can anyone provide me with a proof? That will be very helpful.



Elliptic curves and $GL(2)$ Iwasawa theory
For $n$ large enough the Galois group of $F\_n/F\_{n-1}$ identifies with the kernel $G$ of $\operatorname{GL}\_2(\mathbb{Z}/p^{n+1}\mathbb{Z}) \to \operatorname{GL}\_2(\mathbb{Z}/p^{n}\mathbb{Z})$. Let $P$ and $Q$ be a basis of $E\_{p^{n+1}}$. The norm of $P$ is written as a sum over $a$, $b$, $c$, $d$ modulo $p$ as
\begin{align\*} N\_G(P) &= \sum\_{a,b,c,d} \begin{pmatrix} 1+ap^n & bp^n \\ cp^n & 1+dp^n\end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix}\\
&= \sum\_{a,b,c,d}\bigl( (1+ap^n)P+cp^nQ\bigr)\\
&= p^4P+p^{n+3}(\sum\_{a=0}^{p-1} a)\, P + p^{n+3} (\sum\_{c=0}^{p-1} c)\, Q \\
&= p^4 P\end{align\*}
and similar for $Q$ so it is multiplication by $p^4$ on all of $E\_{p^{n+1}}$.


