Serre's surjective theorem importance.

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 it has? What is its importance?

Solution 1:

For instance, note that $\rho_{\ell}(\mathrm{Gal}(\overline{\Bbb{Q}}/\Bbb{Q}))\cong \mathrm{Gal}(\Bbb{Q}(E[\ell])/\Bbb{Q})$, where $\Bbb{Q}(E[\ell])$ denotes the number field generated by the coordinates of the $\ell$-torsion points of $E$, and $\mathrm{Aut}(E[\ell])\cong \mathrm{GL}_{2}(\mathbb{F}_{\ell})$, so we can solve the inverse Galois problem for $\mathrm{GL}_{2}(\mathbb{F}_{\ell})$.