# Elliptic curves and $GL(2)$ Iwasawa theory

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.

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