Rank distribution in cubic twist families of elliptic curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908409461211136 |
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| author | Ray, Anwesh Shingavekar, Pratiksha |
| author_facet | Ray, Anwesh Shingavekar, Pratiksha |
| contents | Let $a$ be an integer which is not of the form $n^2$ or $-3 n^2$ for $n\in \mathbb{Z}$. Let $E_a$ be the elliptic curve with rational $3$-isogeny defined by $E_a:y^2=x^3+a$, and $K:=\mathbb{Q}(μ_3)$. Assume that the $3$-Selmer group of $E_a$ over $K$ vanishes. It is shown that there is an explicit infinite set of cubefree integers $m$ such that the $3$-Selmer groups over $K$ of $E_{m^2 a}$ and $E_{m^4 a}$ both vanish. In particular, the ranks of these cubic twists are seen to be $0$ over $K$. Our results are proven by studying stability properties of $3$-Selmer groups in cyclic cubic extensions of $K$, via local and global Galois cohomological techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rank distribution in cubic twist families of elliptic curves Ray, Anwesh Shingavekar, Pratiksha Number Theory Algebraic Geometry 11G05, 11R45, 11R34 Let $a$ be an integer which is not of the form $n^2$ or $-3 n^2$ for $n\in \mathbb{Z}$. Let $E_a$ be the elliptic curve with rational $3$-isogeny defined by $E_a:y^2=x^3+a$, and $K:=\mathbb{Q}(μ_3)$. Assume that the $3$-Selmer group of $E_a$ over $K$ vanishes. It is shown that there is an explicit infinite set of cubefree integers $m$ such that the $3$-Selmer groups over $K$ of $E_{m^2 a}$ and $E_{m^4 a}$ both vanish. In particular, the ranks of these cubic twists are seen to be $0$ over $K$. Our results are proven by studying stability properties of $3$-Selmer groups in cyclic cubic extensions of $K$, via local and global Galois cohomological techniques. |
| title | Rank distribution in cubic twist families of elliptic curves |
| topic | Number Theory Algebraic Geometry 11G05, 11R45, 11R34 |
| url | https://arxiv.org/abs/2403.18034 |