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Improved bounds for Serre's open image theorem

Imin Chen, Joshua Swidinsky

TL;DR

This work delivers significantly tighter explicit bounds in Serre's open image theorem for non-CM elliptic curves over $\mathbb{Q}$ under GRH. It refines the Mayle–Wang approach by replacing the deviation group $\delta(G)$ with smaller quotients $\varphi(G)$ in the $2$-adic setting and leverages the Rouse–Zureick-Brown classification of $2$-adic images to bound these objects. The combination with explicit Chebotarev density bounds yields the main result $C_E \le 446\log\mathrm{rad}(2N_E) + 2254$, and analogous improvements hold for quadratic twists. Additionally, the paper proves improved effective isogeny theorems, including twist-sensitive cases, all supported by computational verification. These contributions enhance the practical applicability of Serre’s theorem and deepen understanding of $2$-adic Galois representations of elliptic curves.

Abstract

Let $E$ be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb{Q}$ has open image, and in particular there is a minimal natural number $C_E$ such that the mod $\ell$ representation $\barρ_{E,\ell}$ is surjective for any prime $\ell > C_E$. Assuming the Generalized Riemann Hypothesis, Mayle-Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of $E$ and have explicit constants. The method is based on using effective forms of the Chebotarev density theorem together with the Faltings-Serre method, in particular, using the `deviation group' of the $2$-adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $ρ_{E,2}$ by Rouse and Zureick-Brown, we show in this paper how to further reduce the constants in Mayle-Wang's results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.

Improved bounds for Serre's open image theorem

TL;DR

This work delivers significantly tighter explicit bounds in Serre's open image theorem for non-CM elliptic curves over under GRH. It refines the Mayle–Wang approach by replacing the deviation group with smaller quotients in the -adic setting and leverages the Rouse–Zureick-Brown classification of -adic images to bound these objects. The combination with explicit Chebotarev density bounds yields the main result , and analogous improvements hold for quadratic twists. Additionally, the paper proves improved effective isogeny theorems, including twist-sensitive cases, all supported by computational verification. These contributions enhance the practical applicability of Serre’s theorem and deepen understanding of -adic Galois representations of elliptic curves.

Abstract

Let be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to has open image, and in particular there is a minimal natural number such that the mod representation is surjective for any prime . Assuming the Generalized Riemann Hypothesis, Mayle-Wang gave explicit bounds for which are logarithmic in the conductor of and have explicit constants. The method is based on using effective forms of the Chebotarev density theorem together with the Faltings-Serre method, in particular, using the `deviation group' of the -adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the -adic representation by Rouse and Zureick-Brown, we show in this paper how to further reduce the constants in Mayle-Wang's results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.
Paper Structure (7 sections, 26 theorems, 73 equations, 2 tables)

This paper contains 7 sections, 26 theorems, 73 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Explicit forms of the Chebotarev density theorem
  3. The deviation group $\delta(G)$
  4. The tools of Mayle-Wang
  5. Improved bounds for the effective isogeny theorem
  6. The results of Rouse and Zureick-Brown on $2$-adic images
  7. Improved bounds for Serre's open image theorem

Key Result

Theorem 1.1

serrechebotarev Assume GRH. Let $E$ and $E'$ be two elliptic curves defined over $\mathbb{Q}$. Suppose that $E$ and $E'$ are not $\mathbb{Q}$-isogenous. Then there exists a prime $p$ of good reduction for $E$ and $E'$ such that $a_p(E) \neq a_p(E')$ and satisfying the inequality where $C_1$ is an absolute constant.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1: Chebotarev Density Theorem
  • Theorem 2.2
  • Corollary 2.3
  • ...and 32 more