On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

http://dx.doi.org/10.4153/CMB-2005-002-x
Canad. Math. Bull. 48(2005), 16-31

  Published:2005-03-01
 Printed: Mar 2005

Let $ E $ be an elliptic curve defined over $\Q,$ of conductor $N$ and without complex multiplication. For any positive integer $l$, let $\phi_l$ be the Galois representation associated to the $l$-division points of~$E$. From a celebrated 1972 result of Serre we know that $\phi_l$ is surjective for any sufficiently large prime $l$. In this paper we find conditional and unconditional upper bounds in terms of $N$ for the primes $l$ for which $\phi_l$ is {\emph{not}} surjective.