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# Canonical heights on genus-2 Jacobians

## Title data

Müller, Jan Steffen ; Stoll, Michael:
Canonical heights on genus-2 Jacobians.
In: Algebra & Number Theory. Vol. 10 (2016) Issue 10 . - pp. 2153-2234.
ISSN 1937-0652
DOI: https://doi.org/10.2140/ant.2016.10.2153

## Abstract in another language

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) → ℝ. More specifically, we consider the following two problems, which are important in applications:

1. for a given P ∈ J(K), compute ĥ(P) efficiently;
2. for a given bound B > 0, find all P ∈ J(K) with ĥ(P) ≤ 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.

## Further data

Item Type: Article in a journal Yes canonical height; hyperelliptic curve; curve of genus 2; Jacobian surface; Kummer surface Mathematics Subject Classification Code: 11G50 (11G30 11G10 14G40 14Q05 14G05) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra)Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael StollFacultiesFaculties > Faculty of Mathematics, Physics und Computer ScienceFaculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Yes 500 Science > 510 Mathematics 04 May 2017 09:07 04 May 2017 09:07 https://eref.uni-bayreuth.de/id/eprint/36941