Canonical heights on genus-2 Jacobians

Let $K$ be a number field and let $C ∕ K$ be a curve of genus $2$ with Jacobian variety $J$ . We study the canonical height $ĥ : J (K) \to ℝ$ . More specifically, we consider the following two problems, which are important in applications:

for a given $P \in J (K)$ , compute $ĥ (P)$ efficiently;
for a given bound $B > 0$ , find all $P \in J (K)$ with $ĥ (P) \leq B$ .

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 $h - ĥ$ , which allows a much faster enumeration of the desired set of points.

Our approach is to use the standard decomposition of $h (P) - ĥ (P)$ 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 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.

Keywords

canonical height, hyperelliptic curve, curve of genus 2, Jacobian surface, Kummer surface

Mathematical Subject Classification 2010

Primary: 11G50

Secondary: 11G30, 11G10, 14G40, 14Q05, 14G05

Milestones

Received: 31 March 2016

Revised: 2 August 2016

Accepted: 5 September 2016

Published: 9 December 2016

Authors

Jan Steffen Müller
Institut für Mathematik Carl von Ossietzky Universität Oldenburg D-26111 Oldenburg Germany

Michael Stoll
Mathematisches Institut Universität Bayreuth D-95440 Bayreuth Germany

Vol. 10, No. 10, 2016

Jan Steffen Müller and Michael Stoll

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors