Canonical Heights on Genus Two Jacobians

TL;DR

This paper develops efficient algorithms for computing the canonical height on genus two Jacobians over number fields, improving bounds for height differences and enabling faster enumeration of points with bounded height.

Contribution

It introduces a polynomial-time algorithm for computing canonical heights on genus two Jacobians and refines height bounds to accelerate point enumeration.

Findings

Algorithm runs in polynomial time and is practical.

Improved bounds for height differences enable faster point enumeration.

Replaces prime factorization with coprime factorization for efficiency.

Abstract

Let $K$ be a number field and let $C/K$ be a curve of genus 2 with Jacobian variety $J$. In this paper, we study the canonical height $\hat{h} \colon J(K) \to \mathbb R$. More specifically, we consider the following two problems, which are important in applications: (1) for a given $P \in J(K)$, compute $\hat{h}(P)$ efficiently; (2) for a given bound $B > 0$, find all $P \in J(K)$ with $\hat{h}(P) \le B$. We develop an algorithm running in polynomial time (and fast in practice) to deal with the first problem. Regarding the second problem, we show how one can tweak the naive height $h$ that is usually used to obtain significantly improved bounds for the difference $h - \hat{h}$, which allows a much faster enumeration of the desired set of points. Our approach is to use the standard decomposition of $h(P) - \hat{h}(P)$ as a sum of local `height correction functions'. We study these functions carefully, which leads to efficient ways of computing them and to essentially optimal bounds. To get our polynomial-time algorithm, we have to avoid the factorization step needed to find the finite set of places where the correction might be nonzero. The main innovation at this point is to replace factorization into primes by factorization into coprimes. Most of our results are valid for more general fields with a set of absolute values satisfying the product formula. An analogous approach to (1) above for elliptic curves was used in our recent paper arXiv:1509.08748.