$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

Published 21 Jun 2021 in math.NT | (2106.11141v6)

Abstract: We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell=2$ and $\ell\ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$. For each of these $\ell$, we compute the directed graph of arithmetically maximal $\ell$-power level modular curves $X_H$, compute explicit equations for all but three of them, and classify the rational points on all of them except $X_{\rm ns}^{+}(N)$, for $N = 27, 25, 49, 121$, and two level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$. Aside from the $\ell$-adic images that are known to arise for infinitely many $\bar{\mathbb{Q}}$-isomorphism classes of elliptic curves $E/\mathbb{Q}$, we find only 22 exceptional images that arise for any prime $\ell$ and any $E/\mathbb{Q}$ without complex multiplication; these exceptional images are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\rm ns}^+(\ell)$ with $\ell\ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell$-adic images of Galois for any elliptic curve over $\mathbb{Q}$. In an appendix with John Voight we generalize Ribet's observation that simple abelian varieties attached to newforms on $Γ_1(N)$ are of ${\rm GL}_2$-type; this extends Kolyvagin's theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.

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Summary

The paper presents a complete classification of ℓ-adic Galois images for elliptic curves over ℚ, identifying 22 exceptional subgroups associated with 20 rational j-invariants.
It rigorously enumerates maximal ℓ-power level subgroups and computes explicit modular curves using advanced diophantine and analytic techniques.
The results lead to an efficient algorithm for computing ℓ-adic images, with significant implications for isogeny analysis, torsion growth, and rational point studies.

Essay: $\ell$ -adic Images of Galois for Elliptic Curves over $\mathbb{Q}$ (2106.11141)

Problem Statement and Context

The classification of $\ell$ -adic Galois representations associated to elliptic curves $E/\mathbb{Q}$ is foundational in arithmetic geometry, encoding the arithmetic and geometric structure of torsion points through the action of the absolute Galois group $G_{\mathbb{Q}}$ . Mazur's "Program B" seeks a comprehensive description: given a subgroup $H$ of $GL_2(\widehat{\mathbb{Z}})$ , classify all elliptic curves $E/\mathbb{Q}$ for which $\rho_E(G_{\mathbb{Q}}) \leq H$ .

While for $\ell=2$ and $\ell \geq 13$ the images have been essentially classified, there remained substantial gaps for intermediate primes $\ell=3,5,7,11$ . The paper addresses these cases, producing explicit modular curves, determining their rational points, and demonstrating exceptional images correspond to a strict, finite collection of $j$ -invariants. The outcome is a practical and effective criterion for computing $\ell$ -adic images for any $E/\mathbb{Q}$ .

Classification Framework and Main Results

The $\ell$ -adic image $\rho_{E,\ell^\infty}:G_{\mathbb{Q}}\to GL_2(\mathbb{Z}_\ell)$ is open (finite index) for non-CM curves by Serre’s open image theorem. The central task is identifying which open subgroups arise as images for some $E/\mathbb{Q}$ , which reduces to classifying rational points on corresponding modular curves $X_H$ .

The paper systematically enumerates maximal $\ell$ -power level subgroups $H \leq GL_2(\mathbb{Z}_\ell)$ , for $\ell = 3,5,7,11$ , computes their modular curves $X_H$ , and resolves the diophantine problem of rational points on these curves except for several genus $9$ or higher cases (where analytic rank equals dimension of the Jacobian), and for $X_{\mathrm{ns}}^+(\ell)$ for $\ell \geq 19$ .

Key claims:

Exceptional behavior: For $\ell$ -adic images not arising from infinitely many $j$ -invariants, only 22 conjugacy classes of open subgroups appear for non-CM $E/\mathbb{Q}$ , realized by exactly 20 rational $j$ -invariants.
Completeness conjecture: It is conjectured that these are all exceptional images; any counterexample would necessarily arise from rational points on $X_{\mathrm{ns}}^+(\ell)$ for $\ell\geq 19$ or on the six unresolved higher-genus curves.
Algorithmic implications: This classification enables highly efficient computation of $\ell$ -adic images of Galois for any $E/\mathbb{Q}$ by checking inclusion in known infinite or exceptional cases.
Ribet/Kolyvagin extension: The appendix demonstrates that isogeny factors of modular Jacobians attached to newforms on $\Gamma_1(N)$ are always of $GL_2$ -type, and extends Kolyvagin's rank zero implies algebraic rank zero result to Jacobians of covers $X_H$ .

Technical Innovations and Numerical Outcomes

Subgroup Enumeration & Modular Curves

For each $\ell$ , arithmetically maximal subgroups of $\ell$ -power level—open, with surjective determinant, real points, and modular curves $X_H$ with only finitely many rational $j$ -invariants—are explicitly classified. The paper employs:

Direct enumeration in $GL_2(\ell^{e})$ for sharp upper bounds on level/index/genus.
Explicit equations for most $X_H$ , including high genus models via canonical ring and modular form computations.

Numerical summary:

For $\ell=2,3,5,7,11,13,17,37$ and others, the maximal levels range from 32 to 37, maximal index up to 6655, and maximal genus up to 511.
The vast majority of rational points are cusps or CM; only 22 exceptional images correspond to 20 rational $j$ -invariants.

Rational Point Analysis

Advanced diophantine methods were applied:

Direct Chabauty, Mordell-Weil sieving, analytic rank computations for Jacobians using trace formula, and permutation character techniques for point counting over finite fields.
In most genus 0/1/2/3 cases, explicit models yielded complete rational point enumeration; higher-genus cases remain unresolved, but reasons for the absence of non-CM rational points are provided.

Algorithmic Computation

An efficient algorithm is constructed to compute the $\ell$ -adic image for any given $E/\mathbb{Q}$ :

Only for exceptional $j$ -invariants does $\rho_E$ fall into one of the exceptional images.
For other $j$ , genus 0/1 modular curves parameterize possible images, with explicit models permitting rapid membership testing.

This algorithm has been implemented and applied to large databases, achieving computation of $\ell$ -adic Galois images for all $E/\mathbb{Q}$ of conductor up to $500,000$ for every prime $\ell$ .

Broader Applications and Theoretical Implications

The explicit $\ell$ -adic image classification implies new results for:

Isogeny constraints: For curves with $\ell$ -isogenies, images are as large as allowed by the isogeny, generalizing Greenberg and others for $\ell=3,5,7$ .
Base change and torsion growth: Galois behavior under field extensions and analysis of division fields, with applications to modular curves, torsion points, and entanglement phenomena.
Modular curves and arithmetic geometry: The explicit point counts and higher-genus curve analyses impact approaches to rational points, isolated points on $X_1(N)$ , and gonality bounds.

On the theoretical side, the completeness conjecture, if true, would confirm a robust picture matching the general expectations set by Serre, Mazur, and their successors. The explicit methods shown are scalable, suggest algorithmic approaches can resolve similar classification questions for other arithmetic moduli problems, and provide essential tools for computational number theory.

Future Directions

Resolution of genus $9$ and higher cases, or proof of the completeness conjecture, depends on further advances in high-genus curve arithmetic and analytic rank techniques.
Extension to other fields, base changes, and higher-dimensional analogues (e.g., abelian surfaces).
Deep exploration of Galois cohomology implications, entanglements, and effective image analysis in broader moduli problems.

Conclusion

The paper delivers a detailed classification of $\ell$ -adic Galois images for elliptic curves over $\mathbb{Q}$ for key small primes, establishes a finite list of exceptional cases, conjectures completeness of this list, and provides explicit methods for curve and image determination. This framework bridges computational and theoretical arithmetic geometry, equipping practitioners with algorithmic and structural tools for advanced analysis of Galois representations and their arithmetic consequences.