Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908496637722624 |
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| author | Savoie, Ben |
| author_facet | Savoie, Ben |
| contents | We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny descent, and the existence of infinitely many such curves follows from Tao's constellation theorem for Gaussian primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728 Savoie, Ben Number Theory 11G05 (Primary) 14G05 (Secondary) We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny descent, and the existence of infinitely many such curves follows from Tao's constellation theorem for Gaussian primes. |
| title | Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728 |
| topic | Number Theory 11G05 (Primary) 14G05 (Secondary) |
| url | https://arxiv.org/abs/2506.17605 |