A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points
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| Format: | Preprint |
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2024
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| _version_ | 1866916363681923072 |
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| author | Laga, Jef Swaminathan, Ashvin A. |
| author_facet | Laga, Jef Swaminathan, Ashvin A. |
| contents | Let $\mathcal{F}_g$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over $\mathbb{Q}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_g$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_g$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_09421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points Laga, Jef Swaminathan, Ashvin A. Number Theory Algebraic Geometry 11G30 (Primary) 14G05, 14H25 (Secondary) Let $\mathcal{F}_g$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over $\mathbb{Q}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_g$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_g$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^1$. |
| title | A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points |
| topic | Number Theory Algebraic Geometry 11G30 (Primary) 14G05, 14H25 (Secondary) |
| url | https://arxiv.org/abs/2405.09421 |