New isogenies of elliptic curves over number fields
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918124523094016 |
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| author | Genao, Tyler |
| author_facet | Genao, Tyler |
| contents | Using Galois representations, we analyze fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F$ which has no rational CM, under GRH there exists an effectively computable constant $B:=B(F)\in\mathbb{Z}^+$ such that for any finite extension $L/F$ whose degree $[L:F]$ is coprime to $B$, one has for all elliptic curves $E_{/F}$ that any $L$-rational isogeny on $E$ is $F$-rational. For any number field $F$, under GRH we also prove results for the mod-$\ell$ Galois representations of non-CM elliptic curves with an $F$-rational isogeny of uniformly large prime degree $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05507 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New isogenies of elliptic curves over number fields Genao, Tyler Number Theory 11G05 Using Galois representations, we analyze fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F$ which has no rational CM, under GRH there exists an effectively computable constant $B:=B(F)\in\mathbb{Z}^+$ such that for any finite extension $L/F$ whose degree $[L:F]$ is coprime to $B$, one has for all elliptic curves $E_{/F}$ that any $L$-rational isogeny on $E$ is $F$-rational. For any number field $F$, under GRH we also prove results for the mod-$\ell$ Galois representations of non-CM elliptic curves with an $F$-rational isogeny of uniformly large prime degree $\ell$. |
| title | New isogenies of elliptic curves over number fields |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2405.05507 |