Growth of points on hyperelliptic curves

TL;DR

This paper investigates the growth rate of algebraic points on hyperelliptic curves over $Q$, establishing lower bounds on the number of generated field extensions with fixed degree and bounded discriminant, especially for large degrees.

Contribution

It provides new lower bounds on the number of field extensions generated by points on hyperelliptic curves, extending previous work from elliptic to higher genus curves.

Findings

Number of such extensions grows at least as fast as a power of X.

Growth rate approaches 1/4 as the degree n increases.

Results depend on the genus g and the degree of the defining polynomial.

Abstract

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.