An explicit theory of heights for hyperelliptic Jacobians of genus three

TL;DR

This paper develops explicit equations and height theories for hyperelliptic Jacobians of genus three, enabling better understanding of their arithmetic properties and aiding in Mordell-Weil group computations.

Contribution

It provides explicit equations for the Kummer variety and duplication map of genus three hyperelliptic Jacobians, and develops an explicit height theory over number fields.

Findings

Explicit equations for the Kummer variety in P^7

Explicit duplication map on the Kummer variety

Bound on the difference between naive and canonical height

Abstract

We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field $k$ of characteristic $\neq 2$. 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.