The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915209430433792 |
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| author | Smith, Alexander |
| author_facet | Smith, Alexander |
| contents | Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have $2^{\infty}$-Selmer corank 0 and 50% have $2^{\infty}$-Selmer corank 1. As one consequence, we prove that the Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture. Previously, this result was known by work of the author for elliptic curves over Q satisfying certain technical conditions.
As part of this work, we determine the distribution of 2-Selmer ranks in the quadratic twist family of E. In the cases where this distribution was not already known, it is distinct from the model for distributions of 2-Selmer groups constructed by Poonen and Rains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture Smith, Alexander Number Theory 11G05 Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have $2^{\infty}$-Selmer corank 0 and 50% have $2^{\infty}$-Selmer corank 1. As one consequence, we prove that the Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture. Previously, this result was known by work of the author for elliptic curves over Q satisfying certain technical conditions. As part of this work, we determine the distribution of 2-Selmer ranks in the quadratic twist family of E. In the cases where this distribution was not already known, it is distinct from the model for distributions of 2-Selmer groups constructed by Poonen and Rains. |
| title | The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2503.17619 |