## 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

### MSC:

 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

### Citations:

Zbl 0311.14003; Zbl 0303.14003
Full Text:

### References:

 [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 [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 [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 [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 [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
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