Statistics for 3-isogeny induced Selmer groups of elliptic curves
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910472939241472 |
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| author | Shingavekar, Pratiksha |
| author_facet | Shingavekar, Pratiksha |
| contents | Given a sixth power free integer $a$, let $E_a$ be the elliptic curve defined by $y^2=x^3+a$. We prove explicit results for the lower density of sixth power free integers $a$ for which the $3$-isogeny induced Selmer group of $E_a$ over $\mathbb{Q}(μ_3)$ has dimension $\leq 1$. The results are proven by refining the strategy of Davenport--Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for $3$-isogeny induced Selmer groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Statistics for 3-isogeny induced Selmer groups of elliptic curves Shingavekar, Pratiksha Number Theory 11G05, 11R32, 11R45 Given a sixth power free integer $a$, let $E_a$ be the elliptic curve defined by $y^2=x^3+a$. We prove explicit results for the lower density of sixth power free integers $a$ for which the $3$-isogeny induced Selmer group of $E_a$ over $\mathbb{Q}(μ_3)$ has dimension $\leq 1$. The results are proven by refining the strategy of Davenport--Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for $3$-isogeny induced Selmer groups. |
| title | Statistics for 3-isogeny induced Selmer groups of elliptic curves |
| topic | Number Theory 11G05, 11R32, 11R45 |
| url | https://arxiv.org/abs/2406.03066 |