The possible adelic indices for elliptic curves admitting a rational cyclic isogeny
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908901150031872 |
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| author | Finnerty, Kate Genao, Tyler Mayle, Jacob Rakvi |
| author_facet | Finnerty, Kate Genao, Tyler Mayle, Jacob Rakvi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00652 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The possible adelic indices for elliptic curves admitting a rational cyclic isogeny Finnerty, Kate Genao, Tyler Mayle, Jacob Rakvi Number Theory 11G05, 11F80 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. |
| title | The possible adelic indices for elliptic curves admitting a rational cyclic isogeny |
| topic | Number Theory 11G05, 11F80 |
| url | https://arxiv.org/abs/2512.00652 |