Cyclic isogenies of elliptic curves over fixed quadratic fields
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911459956490240 |
|---|---|
| author | Banwait, Barinder S. Najman, Filip Padurariu, Oana |
| author_facet | Banwait, Barinder S. Najman, Filip Padurariu, Oana |
| contents | Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised.
In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_08891 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cyclic isogenies of elliptic curves over fixed quadratic fields Banwait, Barinder S. Najman, Filip Padurariu, Oana Number Theory 11G05 (primary), 11Y60, 11G15 (secondary) Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest. |
| title | Cyclic isogenies of elliptic curves over fixed quadratic fields |
| topic | Number Theory 11G05 (primary), 11Y60, 11G15 (secondary) |
| url | https://arxiv.org/abs/2206.08891 |