Irrational points on random hyperelliptic curves

Abstract

We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d<g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves have at most $B_d$ points of degree $d$. If $d<g$ is even, we conditionally bound degree $d$ points not pulled back from points of degree $d/2$ on the projective line. Furthermore, we show one may take $B_2=24$ and B_3=114$. Our proofs proceed by refining recent work of Park, which applied tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves.

21 pages. Significantly updated to account for technical hypotheses needed in previous results from the literature; some of our theorems are now conditional