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On the effective version of Serre's open image theorem

Jacob Mayle, Tian Wang

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overlineρ_{E, \ell}$ 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 $\overlineρ_{E, \ell}$ is nonsurjective.

On the effective version of Serre's open image theorem

Abstract

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

Paper Structure

This paper contains 11 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Progress toward Serre's uniformity question
  3. The effective Chebotarev density theorem
  4. An effective isogeny theorem for elliptic curves
  5. Ramified primes
  6. Effective isogeny theorem
  7. A bound on the largest non-surjective prime
  8. Quadratic twists
  9. Completing the proof
  10. An example
  11. Acknowledgments

Key Result

Theorem 1

Assume GRH. If $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then where $\operatorname{rad} n \coloneqq \prod_{p \mid n} p$ denotes the radical of an integer $n$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • Lemma 7
  • proof
  • ...and 15 more