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Computing Canonical Heights
With Little (or No) Factorization

Author: Joseph H. Silverman
Journal: Math. Comp. 66 (1997), 787-805
MSC (1991): Primary 11G05, 11Y50
MathSciNet review: 1388892
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $E/ \mathbb {Q} $ be an elliptic curve with discriminant $\Delta $, and let $P\in E( \mathbb {Q})$. The standard method for computing the canonical height $ \hat h(P)$ is as a sum of local heights $\hat h (P)= \hat \lambda _{\infty }(P)+\sum _{p} \hat \lambda _{p}(P)$. There are well-known series for computing the archimedean height $\hat \lambda _{\infty }(P)$, and the non-archimedean heights $\hat \lambda _{p}(P)$ are easily computed as soon as all prime factors of $\Delta $ have been determined. However, for curves with large coefficients it may be difficult or impossible to factor $\Delta $. In this note we give a method for computing the non-archimedean contribution to $\hat h (P)$ which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.

References [Enhancements On Off] (What's this?)

  • 1. C. Batut, D. Bernardi, H. Cohen, M. Olivier, PARI-GP, Version 1.3.7.
  • 2. D. Bernardi, Décomprime, Version 1.0, un program de décomposition des nombres en facteurs premiers utilisant les courbes elliptiques.
  • 3. H. Cohen, A course in computational algebraic number theory, Graduate Texts in Math., vol. 138, Springer Verlag, Berlin, 1993. MR 94i:11105
  • 4. J. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1992. MR 93m:11053
  • 5. S. Fermiger, Un exemple de courbe elliptique definie sur $\mathbb {Q} $ de rang ${}\ge 19$, C.R. Acad. Sci. Paris 315 (1992), 719-722.
  • 6. H.G. Folz and H.G. Zimmer, What is the rank of the Demjanenko matrix?, J. Symbolic Computation 4 (1987), 53-67. MR 88k:11038
  • 7. F. Hazama, Demjanenko matrix, class numbers, and Hodge group, J. Number Theory 34 (1990), 174-177. MR 90m:11090
  • 8. A. Kraus, Quelques remarques a propos des invariants $c_{4}$, $c_{6}$, et $\Delta $ d'une courbe elliptique, Acta Arith. 54 (1989), 75-80. MR 90j:11045
  • 9. M. Laska, An algorithm for finding a minimal Weierstrass equation for an elliptic curve, Math. Comp. 38 (1982), 257-260. MR 84e:14033
  • 10. H. Lenstra Jr., Factoring integers with elliptic curves, Annals of Math. 126 (1987), 649-673. MR 89g:11125
  • 11. J.-F. Mestre, Construction de courbes elliptique sur $\mathbb {Q} $ de rang ${}\ge 12$, C.R. Acad. Sci. Paris 295 (1982), 643-644. MR 84b:14019
  • 12. -, Un exemple de courbes elliptique definie sur $\mathbb {Q} $ de rang ${}\ge 15$, C.R. Acad. Sci. Paris 314 (1992), 453-455. MR 93b:11071
  • 13. K. Nagao, An example of elliptic curve over $\mathbb {Q} $ with rank ${}\ge 20$, Proc. Japan Acad. 69 (1993), 291-293. MR 95a:11052
  • 14. K. Nagao and T. Kouya, An example of elliptic curve over $\mathbb {Q} $ with rank ${}\ge 21$, Proc. Japan Acad. 70 (1994), 104-105. MR 95e:11063
  • 15. H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1985. MR 88k:11002
  • 16. J. Sands and W. Schwarz, A Demjanenko matrix for abelian fields of prime power conductor, J. Number Theory 52 (1995), 85-97. MR 96b:11141
  • 17. W. Schwarz, Demjanenko matrix and 2-divisibility of class numbers, Arch. Math. 60 (1993), 154-156. MR 94g:11095
  • 18. J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986. MR 87g:11070
  • 19. -, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 151, Springer-Verlag, Berlin and New York, 1994. MR 96b:11074
  • 20. -, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339-358. MR 89d:11049
  • 21. H.M. Tschöpe and H.G. Zimmer, Computation of the Néron-Tate height on elliptic curves, Math. Comp. 48 (1987), 351-370. MR 87m:14025
  • 22. J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular Functions of One Variable IV (B.J. Birch and W.Kuyk, eds.), (Antwerp, 1972, Lect. Notes in Math. 272, Springer-Verlag, Berlin, 1975. MR 52:13850
  • 23. H. Zimmer, A limit formula for the canonical height of an elliptic curve and its application to height computations., Number Theory (R.A. Mollin, ed.), Walter de Gruyter, Berlin-New York, 1990, pp. 641-659. MR 93d:11060

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Additional Information

Joseph H. Silverman
Affiliation: Mathematics Department, Box 1917, Brown University, Providence, Rhode Island 02912

Keywords: Elliptic curve, canonical height
Received by editor(s): October 24, 1995
Additional Notes: Research partially supported by NSF DMS-9424642.
Article copyright: © Copyright 1997 American Mathematical Society

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