On $r$-isogenies over $\mathbb{Q}(ζ_r)$ of elliptic curves with rational $j$-invariants
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914823924613120 |
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| author | Najman, Filip |
| author_facet | Najman, Filip |
| contents | The main goal of this paper is to determine for which prime numbers $r\geq 3$ can an elliptic curve~$E$ defined over $\mathbb Q$ have an $r$-isogeny over $\mathbb Q(ζ_r)$. We study this question under various assumptions on the 2-torsion of $E$. Apart from being a natural question itself, the mod~$r$ representations attached to such $E$ arise in the Darmon program for the generalized Fermat equation of signature $(r,r,p)$, playing a key role in the proof of modularity of certain Frey varieties in the recent work of Billerey, Chen, Dieulefait and Freitas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14131 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $r$-isogenies over $\mathbb{Q}(ζ_r)$ of elliptic curves with rational $j$-invariants Najman, Filip Number Theory The main goal of this paper is to determine for which prime numbers $r\geq 3$ can an elliptic curve~$E$ defined over $\mathbb Q$ have an $r$-isogeny over $\mathbb Q(ζ_r)$. We study this question under various assumptions on the 2-torsion of $E$. Apart from being a natural question itself, the mod~$r$ representations attached to such $E$ arise in the Darmon program for the generalized Fermat equation of signature $(r,r,p)$, playing a key role in the proof of modularity of certain Frey varieties in the recent work of Billerey, Chen, Dieulefait and Freitas. |
| title | On $r$-isogenies over $\mathbb{Q}(ζ_r)$ of elliptic curves with rational $j$-invariants |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.14131 |