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.

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    $\begingroup$ 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. $\endgroup$ – Chris Wuthrich Sep 17 '15 at 8:27
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    $\begingroup$ 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. $\endgroup$ – David Loeffler Sep 17 '15 at 9:02
  • $\begingroup$ @DavidLoeffler assuming user75536 has access to MathSciNet, of course... $\endgroup$ – David Roberts Sep 17 '15 at 10:28
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    $\begingroup$ 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. $\endgroup$ – user75536 Sep 17 '15 at 10:42

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).


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