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arXiv · 2405.09421

A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points

Jef Laga·Ashvin A. Swaminathan
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Abstract

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 $\mathbb{P}^1$.

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Jef Laga, Ashvin A. Swaminathan. 2024-05-15. A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points. https://arxiv.org/abs/2405.09421

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