On the infinitude of elliptic curves over a number field with prescribed small rank
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915791706783744 |
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| author | Zywina, David |
| author_facet | Zywina, David |
| contents | For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the function field $K(T)$. We use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to choose our specializations so that we have control over the bad primes and can perform a $2$-descent to compute ranks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10865 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the infinitude of elliptic curves over a number field with prescribed small rank Zywina, David Number Theory Primary 11G05, Secondary 14J27 For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the function field $K(T)$. We use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to choose our specializations so that we have control over the bad primes and can perform a $2$-descent to compute ranks. |
| title | On the infinitude of elliptic curves over a number field with prescribed small rank |
| topic | Number Theory Primary 11G05, Secondary 14J27 |
| url | https://arxiv.org/abs/2602.10865 |