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Computing canonical heights on elliptic curves in quasi-linear time

Part of: Arithmetic algebraic geometry Computational number theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  26 August 2016

J. Steffen Müller  and
Michael Stoll
J. Steffen Müller
Affiliation:
Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany email jan.steffen.mueller@uni-oldenburg.de
Michael Stoll
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany email Michael.Stoll@uni-bayreuth.de

Abstract

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We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.

MSC classification

Primary: 11G50: Heights
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights 11G05: Elliptic curves over global fields 11Y16: Algorithms; complexity

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