A graph-theoretic approach to computing Selmer groups of elliptic curves $y^2 = x^3 + bx$ over $\mathbb{Q}(i)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915354238779392 |
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| author | Kling, Anthony Savoie, Ben |
| author_facet | Kling, Anthony Savoie, Ben |
| contents | We develop a graph-theoretic algorithm to compute the $φ$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\mathbb{Q}(i)$, where $b \in \mathbb{Z}[i]$ and $φ$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a weighted graph $G_b$, whose vertices are the odd Gaussian primes dividing $b$, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. By applying our algorithm, we explicitly compute the $φ$-Selmer group of $E_b$ when $b$ is a product of inert primes, and we construct several infinite families of elliptic curves over $\mathbb{Q}(i)$ with trivial Mordell-Weil rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22714 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A graph-theoretic approach to computing Selmer groups of elliptic curves $y^2 = x^3 + bx$ over $\mathbb{Q}(i)$ Kling, Anthony Savoie, Ben Number Theory 11G05 (Primary) 14H52, 05C90 (Secondary) We develop a graph-theoretic algorithm to compute the $φ$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\mathbb{Q}(i)$, where $b \in \mathbb{Z}[i]$ and $φ$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a weighted graph $G_b$, whose vertices are the odd Gaussian primes dividing $b$, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. By applying our algorithm, we explicitly compute the $φ$-Selmer group of $E_b$ when $b$ is a product of inert primes, and we construct several infinite families of elliptic curves over $\mathbb{Q}(i)$ with trivial Mordell-Weil rank. |
| title | A graph-theoretic approach to computing Selmer groups of elliptic curves $y^2 = x^3 + bx$ over $\mathbb{Q}(i)$ |
| topic | Number Theory 11G05 (Primary) 14H52, 05C90 (Secondary) |
| url | https://arxiv.org/abs/2410.22714 |