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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
Refereed: Yes
Keywords: canonical height; hyperelliptic curve; curve of genus 2; Jacobian surface; Kummer surface
Subject classification: Mathematics Subject Classification Code: 11G50 (11G30 11G10 14G40 14Q05 14G05)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael Stoll
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 04 May 2017 09:07
Last Modified: 04 May 2017 09:07
URI: https://eref.uni-bayreuth.de/id/eprint/36941