Lower bounds on heights of odd degree points of hyperelliptic curves

TL;DR

This paper establishes lower bounds on the heights of odd degree points on hyperelliptic curves, showing that such points cannot have small Weil height in a density 1 family, using reduction theory for matrix representations.

Contribution

It introduces a reduction theory for SL_n acting on symmetric matrix pairs and applies it to hyperelliptic curves to derive height lower bounds for odd degree points.

Findings

Most odd degree points on hyperelliptic curves have large Weil height.

The results hold for a density 1 family of hyperelliptic curves.

Small height odd degree points are rare or non-existent in the studied family.

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.