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

TL;DR

This paper proves that for hyperelliptic curves of genus at least 4, a positive proportion have no quadratic points beyond those coming from rational points, extending previous results about rational points.

Contribution

It establishes that a positive proportion of monic odd-degree hyperelliptic curves of genus g ≥ 4 lack unexpected quadratic points, generalizing prior work on rational points.

Findings

Positive proportion of curves with no unexpected quadratic points

Extension of Poonen and Stoll's results to quadratic points

Results hold for genus g ≥ 4

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 $\mathbb{P}^1$.