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

Published online by Cambridge University Press:  20 November 2018

Alina Carmen Cojocaru  and
Ernst Kani
Alina Carmen Cojocaru
Affiliation:
The Fields Institute, 222 College Street, Toronto, ON, M5T 3J1 e-mail: alina@fields.utoronto.ca
Ernst Kani
Affiliation:
Queen's University, Department of Mathematics and Statistics, Kingston, ON, K7L 3N6 e-mail: kani@mast.queensu.ca
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Abstract

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Let $E$ be an elliptic curve defined over $\mathbb{Q}$, of conductor $N$ and without complex multiplication. For any positive integer $l$, let ${{\phi }_{1}}$ be the Galois representation associated to the $l$-division points of $E$. From a celebrated 1972 result of Serre we know that ${{\phi }_{1}}$ 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 ${{\phi }_{1}}$ is not surjective.

Keywords

11G0511N3611R45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005 , pp. 16 - 31
Copyright
Copyright © Canadian Mathematical Society 2005

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