A Positive Proportion of Monic Odd-Degree Hyperelliptic Curves of Genus g ≥ 4 Have no Unexpected Quadratic Points

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...

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Bibliographic Details
Published in:International mathematics research notices 2024-10, Vol.2024 (19), p.12857-12866
Main Authors: Laga, Jef, Swaminathan, Ashvin A
Format: Article
Language:English
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Summary: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 $\mathbb{P}^{1}$.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnae184