On the ranks of elliptic curves with isogenies
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2016
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| _version_ | 1866916518678233088 |
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| author | Daniels, Harris B. Goodwillie, Hannah |
| author_facet | Daniels, Harris B. Goodwillie, Hannah |
| contents | In recent years, the question of whether the ranks of elliptic curves defined over $\mathbb{Q}$ are unbounded has garnered much attention. One can create refined versions of this question by restricting one's attention to elliptic curves over $\mathbb{Q}$ with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of elliptic curves over $\mathbb{Q}$ with an $n$-isogeny and rank as large as possible. To do this, we use existing techniques due to Rogers, Rubin, Silverberg, and Nagao and develop a new technique (based on an observation made by Mazur) that is more computationally feasible when the naive heights of the elliptic curves are large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1611_01329 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On the ranks of elliptic curves with isogenies Daniels, Harris B. Goodwillie, Hannah Number Theory In recent years, the question of whether the ranks of elliptic curves defined over $\mathbb{Q}$ are unbounded has garnered much attention. One can create refined versions of this question by restricting one's attention to elliptic curves over $\mathbb{Q}$ with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of elliptic curves over $\mathbb{Q}$ with an $n$-isogeny and rank as large as possible. To do this, we use existing techniques due to Rogers, Rubin, Silverberg, and Nagao and develop a new technique (based on an observation made by Mazur) that is more computationally feasible when the naive heights of the elliptic curves are large. |
| title | On the ranks of elliptic curves with isogenies |
| topic | Number Theory |
| url | https://arxiv.org/abs/1611.01329 |