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

Let K be a number field, $E$ an elliptic curves 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 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.

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.