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arXiv · 2405.10224

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

Jef Laga·Jack A. Thorne
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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.

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Jef Laga, Jack A. Thorne. 2024-05-16. 100% of odd hyperelliptic Jacobians have no rational points of small height. https://arxiv.org/abs/2405.10224

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