Computing canonical heights with little (or no) factorization
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- by Joseph H. Silverman PDF
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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
- C. Batut, D. Bernardi, H. Cohen, M. Olivier, PARI-GP, Version 1.3.7.
- D. Bernardi, Décomprime, Version 1.0, un program de décomposition des nombres en facteurs premiers utilisant les courbes elliptiques.
- Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
- J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
- S. Fermiger, Un exemple de courbe elliptique definie sur $\mathbb {Q}$ de rang ${}\ge 19$, C.R. Acad. Sci. Paris 315 (1992), 719–722.
- H. G. Folz and H. G. Zimmer, What is the rank of the Demjanenko matrix?, J. Symbolic Comput. 4 (1987), no. 1, 53–67. MR 908412, DOI 10.1016/S0747-7171(87)80053-6
- Fumio Hazama, Demjanenko matrix, class number, and Hodge group, J. Number Theory 34 (1990), no. 2, 174–177. MR 1042490, DOI 10.1016/0022-314X(90)90147-J
- Alain Kraus, Quelques remarques à propos des invariants $c_4,\;c_6$ et $\Delta$ d’une courbe elliptique, Acta Arith. 54 (1989), no. 1, 75–80 (French). MR 1024419, DOI 10.4064/aa-54-1-75-80
- Michael Laska, An algorithm for finding a minimal Weierstrass equation for an elliptic curve, Math. Comp. 38 (1982), no. 157, 257–260. MR 637305, DOI 10.1090/S0025-5718-1982-0637305-2
- H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673. MR 916721, DOI 10.2307/1971363
- Jean-François Mestre, Construction d’une courbe elliptique de rang $\geq 12$, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 12, 643–644 (French, with English summary). MR 688896
- Jean-François Mestre, Un exemple de courbe elliptique sur $\textbf {Q}$ de rang $\geq 15$, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 6, 453–455 (French, with English summary). MR 1154385
- Koh-ichi Nagao, An example of elliptic curve over $\textbf {Q}$ with rank $\ge 20$, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 8, 291–293. MR 1249440
- Koh-ichi Nagao and Tomonori Kouya, An example of elliptic curve over $\mathbf Q$ with rank $\geq 21$, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 4, 104–105. MR 1276883
- Hans Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 897531, DOI 10.1007/978-1-4757-1089-2
- Jonathan W. Sands and Wolfgang Schwarz, A Demjanenko matrix for abelian fields of prime power conductor, J. Number Theory 52 (1995), no. 1, 85–97. MR 1331767, DOI 10.1006/jnth.1995.1057
- Wolfgang Schwarz, Demjanenko matrix and $2$-divisibility of class numbers, Arch. Math. (Basel) 60 (1993), no. 2, 154–156. MR 1199672, DOI 10.1007/BF01199101
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
- Joseph H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), no. 183, 339–358. MR 942161, DOI 10.1090/S0025-5718-1988-0942161-4
- Heinz M. Tschöpe and Horst G. Zimmer, Computation of the Néron-Tate height on elliptic curves, Math. Comp. 48 (1987), no. 177, 351–370. MR 866121, DOI 10.1090/S0025-5718-1987-0866121-6
- J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
- Horst G. Zimmer, A limit formula for the canonical height of an elliptic curve and its application to height computations, Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 641–659. MR 1106690
Additional Information
- Joseph H. Silverman
- Affiliation: Mathematics Department, Box 1917, Brown University, Providence, Rhode Island 02912
- MR Author ID: 162205
- ORCID: 0000-0003-3887-3248
- Email: jhs@gauss.math.brown.edu
- Received by editor(s): October 24, 1995
- Additional Notes: Research partially supported by NSF DMS-9424642.
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 787-805
- MSC (1991): Primary 11G05, 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-97-00812-0
- MathSciNet review: 1388892