Elliptic curves and $GL(2)$ Iwasawa theory

Question

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.

Answer 1

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 $E_{p^{n+1}}$.