On the surjectivity of Galois representations associated to elliptic curves over number fields. (13th November 2013) - Larson, Eric; Vaintrob, Dmitry
Abstract : Given an elliptic curve E over a number field K, the ℓ‐torsion points E [ℓ] of E define a Galois representationGal ( K ¯ / K ) → GL 2 ( 픽 ℓ ) . A famous theorem of Serre ( Invent. Math. 15 (1972) 259–331) states that as long as E has no complex multiplication (CM), the mapGal ( K ¯ / K ) → GL 2 ( 픽 ℓ ) is surjective for all but finitely many ℓ. We say that a prime number ℓ 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, the largest exceptional prime of an elliptic curve E without CM is no larger than a constant (depending on K ) times log N E, where N E is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre ( Inst. Hautes Études Sci. Publ. Math. (1981) 323–401).