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

TL;DR

This paper proves Zywina's conjecture on adelic indices for a specific family of non-CM elliptic curves over , using modular curves and recent advances in -adic image analysis.

Contribution

It confirms Zywina's conjecture for non-CM elliptic curves over  with rational cyclic isogenies, extending previous results on Serre's uniformity.

Findings

Zywina's conjecture holds for the specified family.

Analysis of modular curves related to prime isogeny degrees.

Utilization of recent -adic image and isogeny-torsion graph techniques.

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.