The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914948840423424 |
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| author | Chan, Stephanie |
| author_facet | Chan, Stephanie |
| contents | Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $ϕ$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hatϕ:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_ϕ(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hatϕ}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06062 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$ Chan, Stephanie Number Theory 11G05, 11N45, 11R45 Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $ϕ$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hatϕ:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_ϕ(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hatϕ}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$. |
| title | The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$ |
| topic | Number Theory 11G05, 11N45, 11R45 |
| url | https://arxiv.org/abs/2211.06062 |