Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves') says that the image of the representation of $Gal(\bar{K} / K)$ on the $l$-adic Tate module $T\_l(E)$ is open in $GL\_2(\mathbb{Z}\_l)$. 


Is there a modern proof of this written down somewhere using Faltings' Theorem (i.e. the Tate conjecture) or other methods?


Edit: I've just found Ribet's review of Serre's book, which contains fairly detailed sketch of the kind of proof I was after, so I included it below.



Modern proof of Serre's open image theorem?
Here is Ribet's proof (expanding on Ulrich's comment):


Let $G\_K:=Gal(\bar{K} / K)$ and $V\_l:=T\_l(E)\otimes \mathbb{Q}\_l$.


The image $\rho\_{l,E}(G)$ is a closed subgroup of the $l$-adic Lie group $\text{Aut}(V\_l(E)) \cong \text{GL}\_{2}(\mathbb{Q}\_l)$ and is therefore a Lie subgroup of $\text{Aut}(V\_l(E))$. Its Lie algebra $\mathfrak{g}\_l$ is a subalgebra of $\mathfrak{gl}\_{2}(\mathbb{Q}\_l)$. We want to show that $\mathfrak{g}\_l=\mathfrak{gl}(V\_l)\cong \mathfrak{gl}\_2(\mathbb{Q}\_l)$ and the result follows. 


(Note that the Lie algebra of the image $\rho\_{l,E}(G\_K)$ is the tangent space of the identity component of the Zariski closure of $\rho\_{l,E}(G\_K)$ in $\text{GL}\_{2}(\mathbb{Q}\_l)$. So $\mathfrak{g}\_l$ `measures the representation up to finite extensions of the base field $K$', since a finite index subgroup of an algebraic group has the same identity component). 


Now $V\_l$ is irreducible as a $\mathfrak{g}\_l$-module (this is a theorem of Shafarevich, and depends on Siegel's theorem on the finiteness of integral points on curves). Secondly, $\mathfrak{g}\_l$ can't be contained in the subalgebra $\mathfrak{sl}(V\_l)$ of $\mathfrak{gl}(V\_l)$ since $\det(\rho\_{l,E})=\chi\_l$ (where $\chi\_l$ is the cyclotomic character giving the action of Galois on $K^{cycl}$).


This leaves two possibilities for $\mathfrak{g}\_l$: either $\mathfrak{g}\_l$ is $\mathfrak{gl}\_2(\mathbb{Q}\_l)$ and we're done, or $\mathfrak{g}\_l$ is a non-split Cartan subalgebra of $\mathfrak{gl}\_2( \mathbb{Q}\_l)$ (an abelian semisimple algebra coming from a quadratic field extension of $\mathbb{Q}\_l$).


Faltings proved two important facts about represenations $\rho\_{l,E}$:



- $\rho\_{l,E}$ is a semisimple representation of $G\_K$ over $\mathbb{Q}\_l$

- $\text{End}(E)\otimes \mathbb{Q}\_l \cong \text{End}\_{\mathfrak{g}\_l}(V\_l)$.


Faltings results then rule out the possibility that $\mathfrak{g}\_l$ is a non-split Cartan subalgebra of $\mathfrak{gl}\_2( \mathbb{Q}\_l)$ and we're done.


