I'm studying Serre's paper in wich he shows the following theorem:


*Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho\_{\ell}:\mathrm{Gal}(\bar K/K)\longrightarrow\mathrm{Aut}(E[\ell])$$ is surjective for all but finitely many prime numbers $\ell$.*


I see the beauty of this theorem, however what consequence has it? What is its importance?


For example I know that for a non-CM semi-stable elliptic curve $E$ over $\mathbb{Q}$, if the $\ell$-adic representation is surjective then $E[\ell](\mathbb{Q})$ is trivial. 



Serre's surjective theorem importance
There is an obvious application to the inverse Galois problem. And, some other instances of this problem have been extensively studied along the same lines (that is by proving surjectivity of some Galois reps attached to cuspidal modular forms for instance).


