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100% of odd hyperelliptic Jacobians have no rational points of small height
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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.
Forward citations
Cited by 2 Pith papers
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Kummers, spinors, and heights
For odd hyperelliptic Jacobians, an explicit pure-spinor Kummer embedding and duplication map yield a density-one canonical height lower bound matching Lang-Silverman.
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Lower bounds on heights of odd degree points of hyperelliptic curves
For 100% of hyperelliptic curves z^2 = f(x,y) ordered by height, every odd-degree algebraic point of degree at most 2g-1 has Weil height at least (1 + 1/(2g+2) - epsilon) log Ht(f).
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