Galois invariants of the Tate module of an elliptic curve over a number field Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$.
I am trying to understand the proof of proposition I.6.7 in the the book Euler Systems by Rubin (which you can find here : http://swc.math.arizona.edu/aws/1999/99RubinES.pdf)
I think that at some point he uses the fact $T_p(E)^{G_{K_v}}= 0$. Is that true and if yes why ? (we write $T_p(E) = \varprojlim E(\overline{K})[p^n]$, the Tate module of $E$ at $p$).
 A: We’re looking in the local, complete situation above $\ell$ at the $p^m$-torsion points of $E$ for all $m$. What does it mean to say that $T_p(E)^{G_v}\ne0$, where $G_v=G_{K_v}$, the Galois group of an algebraic closure of $K_v$ over $K_v$? It would mean that there was a consistent sequence of $p^m$-torsion points of $E$, in particularly infinitely many of them, that are rational over a finite extension of $K_v$. But since we have the exact sequence
$$
0\>\rightarrow\>\widehat E(\mathfrak m)\>\rightarrow E(\mathfrak o)\>\rightarrow\>\tilde E(\kappa)\rightarrow\>0\,,
$$
this can’t happen. Here, $\widehat E(\mathfrak m)$ is the points of the formal group of $E$ with values in the maximal ideal $\mathfrak m$ of the ring $\mathfrak o$ of $v$-integers of $K_v$ (or some finite extension if necessary); $E(\mathfrak o)$ is the $\mathfrak o$-points of $E$ (same as the $K_v$-points), and $\tilde E(\kappa)$ is the group of points of the reduced curve $\tilde E$ rational over the finite field $\kappa$ of characteristic $\ell$. But the points of the formal group are uniquely divisible by any prime different from $\ell$, so there’s no $p$-torsion there; and there are only finitely many points over the finite field. So no good.
A: $\newcommand{\E}{\overline{E}}$Here's a pretty simple explanation in the good reduction case. Let $\mathcal{E}$ denote the unique elliptic curve model of $E$ over $\mathcal{O}$(:=$\mathcal{O}_{K_v}$) and let $\E$ denote the reduction of $\mathcal{E}$ over the residue field $\mathbb{F}_q$ of $K_v$ (where we denote by $q=\ell^r$ for some $r$). We know then that there is a $G_{K_v}$-equivariant isomorphism $T_p E\cong T_p \E$ where $G_{K_v}$ acts on the latter via its surjection $G_{K_v}\to G_{\mathbb{F}_q}$. A similar statement holds with $T_p$ replaced by $V_p$.
Now, if $T_p \E$ has a Galois fixed point then the matrix representation of of any $\rho(g)$, with $g\in G_{\mathbb{F}_q}$, acting on $V_p \E$ has the form $\begin{pmatrix}1 & \ast\\ 0 & \chi(g)\end{pmatrix}$ for some character $\chi:G_{\mathbb{F}_q}\to \mathbb{Z}_p^\times$. Moreover, since we know that $\det V_p \E$ is isomorphic to the $p$-adic cyclotomic character of $G_{\mathbb{F}_q}$ we deduce that $\chi=\chi_p$. Note then that for any $g\in G_{\mathbb{F}_q}$ we'd have that $\mathrm{tr}\rho(g)=1+\chi_p(g)$. In particular, we get that $\mathrm{tr}\rho(\mathrm{Frob}_{q})=1+q$ where $q$ is the size of the residue field of $\mathcal{O}$. Note though that $\mathrm{tr}\rho(\mathrm{From}_{q})=q+1-\# E(\mathbb{F}_{q})$. So, we deduce that $\# E(\mathbb{F}_q)=0$ which is preposterous. 
It's also clear that this works for additive reduction since this is the same thing as potentially good reduction. 
