An explicit theory of heights for hyperelliptic Jacobians of genus three

Abstract

We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field $k$ of characteristic $\ne 2$. In particular, we provide explicit equations defining the Kummer variety $\mathcal{K}$ as a subvariety of ${\mathbb{P}}^7$, together with explicit polynomials giving the duplication map on $\mathcal{K}$. A careful study of the degenerations of this map then forms the basis for the development of an explicit theory of heights on such Jacobians when $k$ is a number field. We use this input to obtain a good bound on the difference between naive and canonical height, which is a necessary ingredient for the explicit determination of the Mordell-Weil group. We illustrate our results with two examples.

49 pages. v2: mostly minor edits, a couple of small mistakes corrected

References in corpus (1)

• Canonical Heights on Genus Two Jacobians

Cited by in corpus (8)

• The average number of rational points on genus two curves is bounded
• p-adic adelic metrics and Quadratic Chabauty I
• A geometric linear Chabauty comparison theorem
• Computing integral points on hyperelliptic curves using quadratic Chabauty
• Computing Unit Groups of Curves
• Archimedean local height differences on elliptic curves
• On the Diophantine equation $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{m}{l}\right)+d$
• Computing isogenies between Jacobian of curves of genus 2 and 3