A positive proportion of odd degree hyperelliptic curves over $\mathbb{Q}$ have at most 12 pairs of unexpected quadratic points

Joseph Gunther, Jackson S. Morrow

Published 2017-09-07Version 1

We consider hyperelliptic curves $C$ over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. We prove that for $g \geq 4$, a positive proportion have at most 24 quadratic points not obtained by pulling back rational points of the projective line. Our proof proceeds by refining recent innovations of Park on symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves.