Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T02:54:15.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing canonical heights on elliptic curves in quasi-linear time

Part of: Computational number theory Arithmetic problems. Diophantine geometry Arithmetic algebraic 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 [email protected]
Michael Stoll
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 391 - 405
Copyright
© The Author(s) 2016 

References

Bernardi, D., ‘Hauteur p-adique sur les courbes elliptiques’, Seminar on number theory, Paris 1979–1980 , Progress in Mathematics 12 (Birkhäuser, Boston, MA, 1981) 1–14.Google Scholar
Bernstein, D. J., ‘Research announcement: Faster factorization into coprimes’, Preprint, 2004.Google Scholar
Bernstein, D. J., ‘Factoring into coprimes in essentially linear time’, J. Algorithms 54 (2005) 1–30.Google Scholar
Borwein, J. M. and Borwein, P. B., Pi and the AGM , Canadian Mathematical Society Series of Monographs and Advanced Texts 4 (John Wiley & Sons, Inc., New York, 1998).Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265.CrossRefGoogle Scholar
Bost, J.-B. and Mestre, J.-F., Calcul de la hauteur archimédienne des points d’une courbe elliptique par un algorithme quadratiquement convergent et application au calcul de la capacité de l’union de deux intervalles, unpublished manuscript, 1993.Google Scholar
Bradshaw, R. W., ‘Provable computation of motivic $L$ -functions’, PhD Thesis, University of Washington, 2010.Google Scholar
Buchmann, J. A. and Lenstra, H. W. Jr, ‘Approximating rings of integers in number fields’, J. Théor. Nombres Bordeaux 6 (1994) no. 2, 221–260.Google Scholar
Cohen, H., A course in computational algebraic number theory (Springer, Berlin, Heidelberg, New York, 1993).CrossRefGoogle Scholar
Cremona, J., Prickett, M. and Siksek, S., ‘Height difference bounds for elliptic curves over number fields’, J. Number Theory 116 (2006) no. 1, 42–68.Google Scholar
Everest, G. and Ward, T., ‘The canonical height of an algebraic point on an elliptic curve’, New York J. Math. 6 (2000) 331–342.Google Scholar
Flynn, E. V. and Smart, N. P., ‘Canonical heights on the Jacobians of curves of genus 2 and the infinite descent’, Acta Arith. 79 (1997) no. 4, 333–352.CrossRefGoogle Scholar
Néron, A., ‘Quasi-fonctions et hauteurs sur les variétés abéliennes’, Ann. of Math. (2) 82 (1965) 249–331.Google Scholar
Silverman, J. H., ‘Computing heights on elliptic curves’, Math. Comp. 51 (1988) no. 183, 339–358.Google Scholar
Silverman, J. H., Advanced topics in the arithmetic of elliptic curves , Graduate Texts in Mathematics 151 (Springer, New York, 1994).CrossRefGoogle Scholar
Silverman, J. H., ‘Computing canonical heights with little (or no) factorization’, Math. Comp. 66 (1997) no. 218, 787–805.Google Scholar
Tschöpe, H. M. and Zimmer, H. G., ‘Computation of the Néron–Tate height on elliptic curves’, Math. Comp. 48 (1987) no. 177, 351–370.Google Scholar
Zimmer, H. G., ‘A limit formula for the canonical height of an elliptic curve and its application to height computations’, Number theory, Banff, AB (de Gruyter, Berlin, 1990) 641–659.Google Scholar