Computing heights on elliptic curves

Author:
Joseph H. Silverman

Journal:
Math. Comp. **51** (1988), 339-358

MSC:
Primary 11G05; Secondary 11D25, 11Y40, 14G25, 14K15

MathSciNet review:
942161

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Abstract: We describe how to compute the canonical height of points on elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over **R**. We describe a modified version of Tate's series which also converges over **C**, and give an efficient procedure for calculating local heights at non-Archimedean places. In this way we can calculate heights over number fields having complex embeddings. We also give explicit estimates for the tail of our series, and present several examples.

**[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. MR**777279**, 10.1090/S0025-5718-1985-0777279-X**[2]**David A. Cox and Steven Zucker,*Intersection numbers of sections of elliptic surfaces*, Invent. Math.**53**(1979), no. 1, 1–44. MR**538682**, 10.1007/BF01403189**[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). MR**0387292****[4]**Benedict H. Gross,*Local heights on curves*, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 327–339. MR**861983****[5]**Benedict H. Gross and Don B. Zagier,*Heegner points and derivatives of 𝐿-series*, Invent. Math.**84**(1986), no. 2, 225–320. MR**833192**, 10.1007/BF01388809**[6]**Serge Lang,*Elliptic curves: Diophantine analysis*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR**518817****[7]**Serge Lang,*Fundamentals of Diophantine geometry*, Springer-Verlag, New York, 1983. MR**715605****[8]**Michael Laska,*An algorithm for finding a minimal Weierstrass equation for an elliptic curve*, Math. Comp.**38**(1982), no. 157, 257–260. MR**637305**, 10.1090/S0025-5718-1982-0637305-2**[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. MR**704399**, 10.1007/BF01398396**[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. MR**817210****[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. MR**895285**, 10.1515/crll.1987.378.60**[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. MR**0393039****[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. MR**866121**, 10.1090/S0025-5718-1987-0866121-6**[17]**B. L. van der Waerden,*Algebra*, 7th ed., Ungar, New York, 1970.**[18]**Don Zagier,*Large integral points on elliptic curves*, Math. Comp.**48**(1987), no. 177, 425–436. MR**866125**, 10.1090/S0025-5718-1987-0866125-3**[19]**Horst G. Zimmer,*Quasifunctions on elliptic curves over local fields*, J. Reine Angew. Math.**307/308**(1979), 221–246. MR**534221**, 10.1515/crll.1979.307-308.221

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1988-0942161-4

Article copyright:
© Copyright 1988
American Mathematical Society