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Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$

Lorenzo Furio, Davide Lombardo

TL;DR

The paper advances Serre's uniformity question over $\mathbb{Q}$ by proving that for every prime $p>5$ the mod-$p$ Galois image of any non-CM elliptic curve $E/\mathbb{Q}$ is not conjugate to the index-$3$ subgroup $G(p)$ of the normaliser of a non-split Cartan. It develops an effective surjectivity framework, refines local analyses of inertia and canonical subgroups, and conducts a detailed study of modular units on the modular curve $X_{G(p)}$ to bound $|j(E)|$ (via $|\log|q||$). A combination of explicit height comparisons, local constraints, and Abel-summation-based refinements yields $|\log|q||<39$ and an explicit prime bound $p<103{,}000$ in the remaining cases, enabling a finite computational verification that excludes the $G(p)$-image for all $p>5$. Consequently, the only possible non-surjective image for primes $p>37$ remains the normaliser of a Cartan, with the exceptional case $p=5$ already observed in prior work. Together, these results substantially advance the uniformity program by reducing the problem to a finite check and clarifying the landscape of residual images for large primes.

Abstract

Serre's uniformity question asks whether there exists a bound $N>0$ such that, for every non-CM elliptic curve $E$ over $\mathbb{Q}$ and every prime $p>N$, the residual Galois representation $ρ_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{Aut}(E[p])$ is surjective. The work of many authors has shown that, for $p>37$, this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup $C_{ns}^+(p)$. Zywina has further proved that, whenever $ρ_{E,p}$ is not surjective for $p>37$, its image is either $C_{ns}^+(p)$ or a certain subgroup $G(p)$ of $C_{ns}^+(p)$ of index $3$. Recently, Le Fourn and Lemos showed that the index-$3$ case cannot arise for $p>1.4 \cdot 10^7$. We strengthen this result by proving that the image of $ρ_{E, p}$ is not conjugate to $G(p)$ for any prime larger than $5$.

Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$

TL;DR

The paper advances Serre's uniformity question over by proving that for every prime the mod- Galois image of any non-CM elliptic curve is not conjugate to the index- subgroup of the normaliser of a non-split Cartan. It develops an effective surjectivity framework, refines local analyses of inertia and canonical subgroups, and conducts a detailed study of modular units on the modular curve to bound (via ). A combination of explicit height comparisons, local constraints, and Abel-summation-based refinements yields and an explicit prime bound in the remaining cases, enabling a finite computational verification that excludes the -image for all . Consequently, the only possible non-surjective image for primes remains the normaliser of a Cartan, with the exceptional case already observed in prior work. Together, these results substantially advance the uniformity program by reducing the problem to a finite check and clarifying the landscape of residual images for large primes.

Abstract

Serre's uniformity question asks whether there exists a bound such that, for every non-CM elliptic curve over and every prime , the residual Galois representation is surjective. The work of many authors has shown that, for , this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup . Zywina has further proved that, whenever is not surjective for , its image is either or a certain subgroup of of index . Recently, Le Fourn and Lemos showed that the index- case cannot arise for . We strengthen this result by proving that the image of is not conjugate to for any prime larger than .
Paper Structure (15 sections, 44 theorems, 139 equations)

This paper contains 15 sections, 44 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries
  4. Faltings height and modular height of rational elliptic curves
  5. Local analysis
  6. The image of the inertia subgroups
  7. The canonical subgroup
  8. An effective surjectivity theorem
  9. $K=\mathbb{Q}$
  10. $K$ is a generic number field
  11. Modular units and an upper bound on $\log|q|$
  12. Modular Units
  13. Comparing $\log|q|$ with $p$
  14. Abel summation and a sharper bound on $\log|q|$
  15. Conclusion of the proof of Theorems \ref{['solution']} and \ref{['solution for small primes']}

Key Result

Theorem 1.2

Suppose that $\rho_{E,p}$ is not surjective for a non-CM elliptic curve $\faktor{E}{\mathbb{Q}}$ and a prime $p>37$.

Theorems & Definitions (109)

  • Theorem 1.2: Zywina
  • Theorem 1.3: Le Fourn, Lemos lefournlemos21
  • Claim 1.4: Bajolet, Bilu, Matschke
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1: Bilu, Parent, Rebolledo
  • Lemma 2.2
  • proof
  • ...and 99 more