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arXiv · 2512.00652

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

Kate Finnerty·Tyler Genao·Jacob Mayle·Rakvi
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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.

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Kate Finnerty, Tyler Genao, Jacob Mayle, Rakvi. 2025-11-29. The possible adelic indices for elliptic curves admitting a rational cyclic isogeny. https://arxiv.org/abs/2512.00652

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