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

Published:2005-03-01
Printed: Mar 2005
• Alina Carmen Cojocaru
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## Abstract

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.
 MSC Classifications: 11G05 - Elliptic curves over global fields [See also 14H52] 11N36 - Applications of sieve methods 11R45 - Density theorems