Let $K$ be a number field, and let $A$ and $B$ be two elliptic curves over $K$. For a nonzero prime ideal $\mathfrak{l}$ of $K$, outside a finite set of primes, let $a_\mathfrak{l}$ and $b_\mathfrak{l}$ be the usual error terms $N(\mathfrak{l})+1-\bar A_\mathfrak{l}(\mathcal{O}/\mathfrak{l})$ and $N(\mathfrak{l})+1-\bar B_\mathfrak{l}(\mathcal{O}/\mathfrak{l})$ respectively (here $N$ is the absolute norm and $\bar X_\mathfrak{l}$ denotes the reduction of the curve $X$ at a good place $\mathfrak{l}$).

It is well known that the collection of these error terms determines the $K$-isogeny class of the curve considered. I am at the moment missing how the "correct" proof of this fact goes.

What I see so far is that, thanks to Faltings' Isogeny Theorem, it is enough to show that ${\rm Hom}_{G_K}(T_p(A),T_p(B))$ is non-zero, for some prime $p$. Now, our assumption that the packages of error terms of $A$ and $B$ coincide for almost all primes of $K$ ensures that the $G_K$-representations $T_p(A)\otimes\mathbf{Q}_p$ and $T_p(B)\otimes\mathbf{Q}_p$ are ${\it locally}$ isomorphic for almost all primes of $K$. How do we get then that they are ${\it globally}$ isomorphic, and hence conclude

${\rm Hom}_{G_K}(T_p(A),T_p(B))\neq 0$? If they came from an automorphic form, I see that one can use the strong multiplicity one result by J-L. But otherwise how can one complete the argument?

Abelian l-adic representations and elliptic curves, IV-15, Proposition. $\endgroup$ – ACL Nov 26 '11 at 14:15