Given an elliptic curve E over a number field K, the β-torsion
points E[β] of E define a Galois representation \gal(\bar{K}/K) \to
\gl_2(\ff_\ell). A famous theorem of Serre states that as long as E has no
Complex Multiplication (CM), the map \gal(\bar{K}/K) \to \gl_2(\ff_\ell) is
surjective for all but finitely many β.
We say that a prime number β is exceptional (relative to the pair
(E,K)) if this map is not surjective. Here we give a new bound on the largest
exceptional prime, as well as on the product of all exceptional primes of E.
We show in particular that conditionally on the Generalized Riemann Hypothesis
(GRH), the largest exceptional prime of an elliptic curve E without CM is no
larger than a constant (depending on K) times logNEβ, where NEβ is the
absolute value of the norm of the conductor. This answers affirmatively a
question of Serre