Generalized class group actions on oriented elliptic curves with level structure
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929745507123200 |
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| author | Arpin, Sarah Castryck, Wouter Eriksen, Jonathan Komada Lorenzon, Gioella Vercauteren, Frederik |
| author_facet | Arpin, Sarah Castryck, Wouter Eriksen, Jonathan Komada Lorenzon, Gioella Vercauteren, Frederik |
| contents | We study a large family of generalized class groups of imaginary quadratic orders $O$ and prove that they act freely and (essentially) transitively on the set of primitively $O$-oriented elliptic curves over a field $k$ (assuming this set is non-empty) equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder $O' \subseteq O$ on the set of $O'$-oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized class group actions on oriented elliptic curves with level structure Arpin, Sarah Castryck, Wouter Eriksen, Jonathan Komada Lorenzon, Gioella Vercauteren, Frederik Number Theory We study a large family of generalized class groups of imaginary quadratic orders $O$ and prove that they act freely and (essentially) transitively on the set of primitively $O$-oriented elliptic curves over a field $k$ (assuming this set is non-empty) equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder $O' \subseteq O$ on the set of $O'$-oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems. |
| title | Generalized class group actions on oriented elliptic curves with level structure |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.14450 |