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arXiv · 2507.08652

Lower bounds on heights of odd degree points of hyperelliptic curves

Jef Laga·Jack A. Thorne
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Abstract

We develop a reduction theory for the representation of $\mathrm{SL}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show that, in a density $1$ family, an odd degree point $P$ of degree at most $2g-1$ on the hyperelliptic curve $z^2 = f_0x^{2g+2} + f_1 x^{2g+1} y + \cdots + f_{2g+2}y^{2g+2}$ cannot have small Weil height.

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Jef Laga, Jack A. Thorne. 2025-07-11. Lower bounds on heights of odd degree points of hyperelliptic curves. https://arxiv.org/abs/2507.08652

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