Growth of points on hyperelliptic curves

Fix a hyperelliptic curve $C/\mathbb{Q}$ of genus $g$, and consider the number fields $K/\mathbb{Q}$ generated by the algebraic points of $C$. In this paper, we study the number of such extensions with fixed degree $n$ and discriminant bounded by $X$. We show that when $g \geq 1$ and $n$ is sufficiently large relative to the degree of $C$, with $n$ even if the degree of the defining polynomial of $C$ is even, there are $\gg X^{c_n}$ such extensions, where $c_n$ is a positive constant depending on $g$ which tends to $1/4$ as $n \to \infty$. This result builds on work of Lemke Oliver and Thorne who, in the case where $C$ is an elliptic curve, put lower bounds on the number of extensions with fixed degree and bounded discriminant over which the rank of $C$ grows with specified root number.