Kummers, spinors, and heights

Abstract

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 $|2\hat{I}˜|$ 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 $100\%$ of monic, degree $2g+1$ polynomials $f(x)\in \mathbb{Z}[x]$ of nonzero discriminant $\hat{I}”(f)$ have the property that, for any non-trivial point $P\in J(\mathbb{Q})$, the canonical height of $P$ satisfies ${\hat{h}}_{\hat{I}}˜(P)\ge \left(\frac{3g-1}{4g(2g+1)}-\hat{I}\mu \right)\log |\hat{I}”(f)|$. This is a `density 1' form of the Lang--Silverman conjecture.