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

We study the universal family of odd hyperelliptic curves of genus $g\ge 1$ 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 ${\mathrm{S}\mathrm{O}}_{2g+1}$ 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.