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

[摘要] Let $ E $ be an elliptic curve defined over$Q,$ of conductor $N$ and without complex multiplication. For anypositive integer $l$, let $phi_l$ be the Galois representationassociated to the $l$-division points of~$E$. From a celebrated1972 result of Serre we know that $phi_l$ is surjective for anysufficiently large prime $l$. In this paper we find conditionaland unconditional upper bounds in terms of $N$ for the primes $l$for which $phi_l$ is {emph{not}} surjective.

[学科分类] 数学（综合）

[关键词] Yang-Mills connection;vector bundle;gauge transformation