Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation ${\overline{\rho }}_{E,\ell }$ of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime $\ell$, linear in the logarithm of the conductor of $E$, such that ${\overline{\rho }}_{E,\ell }$ is nonsurjective.

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关于塞尔开像定理的有效版本

让$乙/\mathbb{问}$是没有复杂乘法的椭圆曲线。根据 Serre 开像定理，mod$\ell$伽罗瓦表示法${\stackrel{￣}{\rho }}_{乙,\ell }$的$乙$对于每个素数都是满射$\ell$足够大。在广义黎曼假设下，我们给出最大素数的明确上限$\ell$，与导体的对数呈线性关系$乙$，使得${\stackrel{￣}{\rho }}_{乙,\ell }$是非满射的。