The average Mordell-Weil rank of elliptic surfaces over number fields
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866918228701216768 |
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| author | Kloosterman, Remke |
| author_facet | Kloosterman, Remke |
| contents | Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside $|\mathcal{B}|$ where the Mordell-Weil rank is at least $r+1$ is a sparse subset.
In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over $\mathbb{Q}$ and prove a similar result for elliptic surfaces over arbitrary number fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_12102 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The average Mordell-Weil rank of elliptic surfaces over number fields Kloosterman, Remke Number Theory Algebraic Geometry Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside $|\mathcal{B}|$ where the Mordell-Weil rank is at least $r+1$ is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over $\mathbb{Q}$ and prove a similar result for elliptic surfaces over arbitrary number fields. |
| title | The average Mordell-Weil rank of elliptic surfaces over number fields |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2204.12102 |