zbMATH — the first resource for mathematics

Computig heights on elliptic curves. (English) Zbl 0656.14016
An efficient algorithm for computing the global canonical height \(\hat h\) on the group of rational points E(K) of an elliptic curve E over a number field K is given. The algorithm proceeds by calculating local canonical heights and adding them up with appropriate multiplicities.
At non-Archimedean places v of K, the local canonical height \(\hat h_ v\) depends on the reduction behaviour of E modulo v, and explicit formulae can be given for \(\hat h_ v\), where the one holding in the case of multiplicative reduction modulo v goes back already to the author’s dissertation.
At Archimedean places v of K, a fast-converging series, due to Tate in case K has real completion \(K_ v\cong {\mathbb{R}}\) and modified by the author in case K has complex completion \(K_ v\cong {\mathbb{C}}\), is used for calculating the local height \(\hat h_ v.\)
Tate’s series actually is independent of whether or not the place v of K is Archimedean or non-Archimedean. The author derives an error estimate for the tail of Tate’s series which is similar to the one obtained by the reviewer [“A limit formula for the canonical height on an elliptic curve and its application to height computations” (to appear in Proc. First Conf. CNTA, Banff, Canada 1988)]. The error estimate is based on a resultant result (proposition 3.1 on page 347) of the same type as the one in an earlier paper of the reviewer [Math. Z. 147, 35-51 (1976; Zbl 0303.14003), formulas (7) and (8) on page 44].
Reviewer: H.G.Zimmer

14H25 Arithmetic ground fields for curves
14G05 Rational points
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
11D25 Cubic and quartic Diophantine equations
14H52 Elliptic curves
Full Text: DOI
[1] Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), no. 170, 473 – 481. · Zbl 0606.14021
[2] David A. Cox and Steven Zucker, Intersection numbers of sections of elliptic surfaces, Invent. Math. 53 (1979), no. 1, 1 – 44. · Zbl 0444.14004 · doi:10.1007/BF01403189 · doi.org
[3] P. Deligne, Courbes elliptiques: formulaire d’après J. Tate, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 53 – 73. Lecture Notes in Math., Vol. 476 (French). · Zbl 1214.11075
[4] Benedict H. Gross, Local heights on curves, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 327 – 339.
[5] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of \?-series, Invent. Math. 84 (1986), no. 2, 225 – 320. · Zbl 0608.14019 · doi:10.1007/BF01388809 · doi.org
[6] Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. · Zbl 0388.10001
[7] Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. · Zbl 0528.14013
[8] Michael Laska, An algorithm for finding a minimal Weierstrass equation for an elliptic curve, Math. Comp. 38 (1982), no. 157, 257 – 260. · Zbl 0493.14016
[9] D. W. Masser and G. Wüstholz, Fields of large transcendence degree generated by values of elliptic functions, Invent. Math. 72 (1983), no. 3, 407 – 464. · Zbl 0516.10027 · doi:10.1007/BF01398396 · doi.org
[10] J. H. Silverman, The Néron-Tate Height on Elliptic Curves, Ph.D. thesis, Harvard, 1981.
[11] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[12] Joseph H. Silverman, A quantitative version of Siegel’s theorem: integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60 – 100. · Zbl 0608.14021 · doi:10.1515/crll.1987.378.60 · doi.org
[13] J. H. Silverman, Elliptic Curve Calculator v. 5.05, a program for the Apple Macintosh computer, 1987.
[14] 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) Springer, Berlin, 1975, pp. 33 – 52. Lecture Notes in Math., Vol. 476. · Zbl 1214.14020
[15] J. T. Tate, Letter to J.-P. Serre, Oct. 1, 1979.
[16] Heinz M. Tschöpe and Horst G. Zimmer, Computation of the Néron-Tate height on elliptic curves, Math. Comp. 48 (1987), no. 177, 351 – 370. · Zbl 0611.14028
[17] B. L. van der Waerden, Algebra, 7th ed., Ungar, New York, 1970. · Zbl 0137.25403
[18] Don Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), no. 177, 425 – 436. · Zbl 0611.10008
[19] Horst G. Zimmer, Quasifunctions on elliptic curves over local fields, J. Reine Angew. Math. 307/308 (1979), 221 – 246. · Zbl 0399.14014 · doi:10.1515/crll.1979.307-308.221 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.