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arXiv · 2109.08656

On the effective version of Serre's open image theorem

Jacob Mayle·Tian Wang
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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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Jacob Mayle, Tian Wang. 2021-09-17. On the effective version of Serre's open image theorem. https://doi.org/10.1112/blms.13002

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