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The possible adelic indices for elliptic curves admitting a rational cyclic isogeny

Kate Finnerty, Tyler Genao, Jacob Mayle, Rakvi

TL;DR

<p>We prove Zywina's conjecture on adelic indices for non-CM elliptic curves over Q that admit a rational cyclic isogeny by performing a prime-wise, modular-curve–driven classification of adelic Galois images. The method combines twists, level-lowering techniques, and extensive rational-point computations (Chabauty, quadratic Chabauty, Mordell–Weil sieve) on a web of fiber products of modular curves X_G, to bound ell-adic images for primes ell  37 and to explicitly determine admissible adelic indices for primes p in {2,3,5,7,11,13,17,37}. A product formula for the adelic index is established, expressing [GL_2(f Z):G_E] as a product of local commutator indices and an entanglement factor, enabling reduction to finite data. For each p in {11,17,37} we exhibit the finite sets I_p of possible indices, and for p in {2,3,5,7,13} we provide explicit lattices and rational points verifying that all resulting indices lie in the conjectured finite set I. The results confirm Zywina’s conjecture for the specified family and strengthen Serre’s uniformity picture for curves with rational isogenies, supplying a detailed computational framework and catalog of concrete examples.</p>

Abstract

In the 1970s, Serre proved that the adelic index of a non-CM elliptic curve over a number field is finite. More recently, Zywina conjectured the complete set of adelic indices for such curves over $\mathbb{Q}$. In this article, we prove that Zywina's conjecture is true for the family of non-CM elliptic curves over $\mathbb{Q}$ that admit a nontrivial rational cyclic isogeny. This strengthens a result of Lemos that resolved Serre's uniformity question for the same family of curves. Our proof proceeds by analyzing a collection of modular curves associated with each prime isogeny degree, using recent advances on $\ell$-adic images, isogeny-torsion graphs, and computations of models and rational points.

The possible adelic indices for elliptic curves admitting a rational cyclic isogeny

TL;DR

<p>We prove Zywina's conjecture on adelic indices for non-CM elliptic curves over Q that admit a rational cyclic isogeny by performing a prime-wise, modular-curve–driven classification of adelic Galois images. The method combines twists, level-lowering techniques, and extensive rational-point computations (Chabauty, quadratic Chabauty, Mordell–Weil sieve) on a web of fiber products of modular curves X_G, to bound ell-adic images for primes ell  37 and to explicitly determine admissible adelic indices for primes p in {2,3,5,7,11,13,17,37}. A product formula for the adelic index is established, expressing [GL_2(f Z):G_E] as a product of local commutator indices and an entanglement factor, enabling reduction to finite data. For each p in {11,17,37} we exhibit the finite sets I_p of possible indices, and for p in {2,3,5,7,13} we provide explicit lattices and rational points verifying that all resulting indices lie in the conjectured finite set I. The results confirm Zywina’s conjecture for the specified family and strengthen Serre’s uniformity picture for curves with rational isogenies, supplying a detailed computational framework and catalog of concrete examples.</p>

Abstract

In the 1970s, Serre proved that the adelic index of a non-CM elliptic curve over a number field is finite. More recently, Zywina conjectured the complete set of adelic indices for such curves over . In this article, we prove that Zywina's conjecture is true for the family of non-CM elliptic curves over that admit a nontrivial rational cyclic isogeny. This strengthens a result of Lemos that resolved Serre's uniformity question for the same family of curves. Our proof proceeds by analyzing a collection of modular curves associated with each prime isogeny degree, using recent advances on -adic images, isogeny-torsion graphs, and computations of models and rational points.

Paper Structure

This paper contains 28 sections, 24 theorems, 78 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Galois Representations
  5. Isogenies
  6. Twists
  7. Level Results
  8. Modular Curves
  9. Rational Point Computations
  10. Map to a Rank 0 Elliptic Curve
  11. Sieving with Respect to a Projection
  12. The Method of Chabauty and Coleman
  13. Quadratic Chabauty and the Mordell--Weil Sieve
  14. Possible ell-adic Images
  15. Strategy and Preliminary Results
  16. ...and 13 more sections

Key Result

Theorem 1

Let $E$ be an elliptic curve over a number field. If $E$ is non-CM, then $G_E$ is an open subgroup of $\mathop{\mathrm{GL}}\nolimits_2(\widehat{\mathbb{Z}})$, equivalently of finite index in $\mathop{\mathrm{GL}}\nolimits_2(\widehat{\mathbb{Z}})$.

Figures (5)

  • Figure 1: The lattice $\mathcal{L}_5(X_0(5))$
  • Figure 2: The lattice $\mathcal{L}_5(X_{\mathrm{sp}}(5))$
  • Figure 3: The lattice $\mathcal{L}_5(X_0(25))$
  • Figure 4: The lattice $\mathcal{L}_7(X_0(7))$
  • Figure 5: The lattice $\mathcal{L}_{13}(X_0(13))$

Theorems & Definitions (39)

  • Theorem 1: Serre's Open Image Theorem, Ser72
  • Conjecture 2: Conjecture 1.5, ZywinaOpen
  • Theorem 3: Theorem 1.1, MR3885140
  • Theorem 4
  • Definition 5
  • Theorem 6: Theorem 1, MR675184
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9: Corollary 2.3, zywina2022possibleindicesgaloisimage
  • ...and 29 more