Kummers, spinors, and heights

Let $f(x)=x^{2g+1}+c_1{x}^{2g}+\cdots +c_{2g+1}\in k[x]$ 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 $\mathrm{\mid }2\mathrm{\Theta }\mathrm{\mid }$ may be described explicitly, for any $g\ge 1$, 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 $100\mathrm{\%}$ of monic, degree $2g+1$ polynomials $f(x)\in \mathbb{Z}[x]$ of nonzero discriminant $\mathrm{\Delta }(f)$ have the property that, for any non-trivial point $P\in J(\mathbb{Q})$, the canonical height of $P$ satisfies ${\widehat{h}}_{\mathrm{\Theta }}(P)\ge (\frac{3g-1}{4g(2g+1)}-\epsilon )\log \mathrm{\mid }\mathrm{\Delta }(f)\mathrm{\mid }$. This is a `density 1’ form of the Lang–Silverman conjecture.