A positive proportion of monic odd-degree hyperelliptic curves of genus $g\ge 4$ 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 $g\ge 3$, 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 $g\ge 4$, 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 ${\mathbb{P}}^1$.

Accepted version