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

Let E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l , let ϕ l be the Galois representation associated to the l -division points of E . From a celebrated 1972 result of Serre we know that ϕ 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 ϕ l is not surjective.