On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields

Details On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields

2012-03-30

arxiv.org/abs/1204.0046v1

Mathematics

Number Theory

Scientific paper

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ 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 $\ell$. We say that a prime number $\ell$ 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 $\log N_E$, where $N_E$ is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre.

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