We study the universal family of odd hyperelliptic curves of genus g ≥ 1 over Q. We relate the heights of Q-points of Jacobians of curves in this family to the reduction theory of the representation of SO2g+1 on self-adjoint (2g + 1) ×(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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Laga et al. (2024) studied this question.
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