On the effective version of Serre's open image theorem

Abstract

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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• Explicit Large Image Theorems for Modular Forms
• Explicit isogenies of prime degree over quadratic fields
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