100% of odd hyperelliptic Jacobians have no rational points of small height
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| Format: | Preprint |
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2024
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| _version_ | 1866911878847922176 |
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| author | Laga, Jef Thorne, Jack A. |
| author_facet | Laga, Jef Thorne, Jack A. |
| contents | We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of $\mathrm{SO}_{2g+1}$ on self-adjoint $(2g + 1) \times(2g + 1)$-matrices. Using this theory, we show that in a density 1 subset, the Jacobians of these curves have no nontrivial rational points of small height. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 100% of odd hyperelliptic Jacobians have no rational points of small height Laga, Jef Thorne, Jack A. Number Theory We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of $\mathrm{SO}_{2g+1}$ on self-adjoint $(2g + 1) \times(2g + 1)$-matrices. Using this theory, we show that in a density 1 subset, the Jacobians of these curves have no nontrivial rational points of small height. |
| title | 100% of odd hyperelliptic Jacobians have no rational points of small height |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.10224 |