Kummers, spinors, and heights

TL;DR

This paper explicitly describes the embedding of Jacobians of odd hyperelliptic curves into projective space using spinor theory and applies this to establish lower bounds on heights of rational points, supporting a density version of the Lang--Silverman conjecture.

Contribution

It provides an explicit description of the Jacobian embedding via pure spinors and connects this to height bounds on rational points, advancing understanding of the Lang--Silverman conjecture.

Findings

Explicit embedding of Jacobians using pure spinors.

Lower bounds on heights of rational points related to polynomial discriminant.

Density 1 result supporting the Lang--Silverman conjecture.

Abstract

Let $f(x) = x^{2g+1} + c_1 x^{2g} + \dots + c_{2g+1} \in k[x]$ be a polynomial of nonzero discriminant, and let $J$ denote the Jacobian of the odd hyperelliptic curve $C : y^2 = f(x)$. We show that the morphism $J \to \mathbb{P}^{2^g-1}$ associated to the linear system $|2 \Theta|$ may be described explicitly, for any $g \geq 1$, using the theory of pure spinors. We apply this theory to study the heights of rational points in $J(k)$, when $k$ is a number field. As a particular consequence, we show that $100\%$ of monic, degree $2g+1$ polynomials $f(x) \in \mathbb{Z}[x]$ of nonzero discriminant $\Delta(f)$ have the property that, for any non-trivial point $P \in J(\mathbb{Q})$, the canonical height of $P$ satisfies $ \widehat{h}_\Theta(P) \geq \left(\frac{3g-1}{4g(2g+1)} - \epsilon\right) \log | \Delta(f) |$. This is a `density 1' form of the Lang--Silverman conjecture.