100% of odd hyperelliptic Jacobians have no rational points of small height

TL;DR

This paper proves that for a large class of odd hyperelliptic Jacobians over rationals, there are no rational points of small height, linking height properties to reduction theory of certain algebraic representations.

Contribution

It establishes a density 1 result showing that most odd hyperelliptic Jacobians lack small-height rational points, using a novel connection to the reduction theory of SO(2g+1) representations.

Findings

Most odd hyperelliptic Jacobians have no rational points of small height.

The height distribution of rational points relates to the reduction theory of specific algebraic groups.

A density 1 subset of these Jacobians exhibits this property.

Abstract

We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of $\mathrm{SO}_{2g+1}$ on self-adjoint $(2g + 1) \times(2g + 1)$-matrices. Using this theory, we show that in a density 1 subset, the Jacobians of these curves have no nontrivial rational points of small height.