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

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 $\mathbb{P}^{1}$.