### References & Citations

# Mathematics > Number Theory

# Title: On the effective version of Serre's open image theorem

(Submitted on 17 Sep 2021)

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Under GRH, we give an upper bound on the largest non-surjective prime, in terms of the conductor of $E$, that makes effective a bound of the same asymptotic quality due to Larson-Vaintrob.

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