Kummers, spinors, and heights

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.