# 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 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.

• Equivalently: The Galois group of $K(E[\ell])$ is isomorphic to $\operatorname{GL}_2(\mathbb{F}_{\ell})$. This field turns up when one does an $\ell$-descent, or more generally when one studies the $\ell$-Selmer group. For instance for an Euler system it is great to have a large Galois group there, for an explicit $\ell$-descent it is rather the opposite. – Chris Wuthrich Sep 17 '15 at 8:27
• A great way of learning about how a theorem is used is to look up on MathSciNet which papers (and which reviews) cite the original paper, and skim through a few of them to see how the result you're interested in gets applied. – David Loeffler Sep 17 '15 at 9:02
• @DavidLoeffler assuming user75536 has access to MathSciNet, of course... – David Roberts Sep 17 '15 at 10:28
• Unfortunately I don't have access to MathSciNet. Before post this question I spent two weeks reading papers found in internet relate to this theorem. However none of them help me to understand the importance of the Serre's paper. – user75536 Sep 17 '15 at 10:42