Computation of the Néron-Tate height on elliptic curves

Authors:
Heinz M. Tschöpe and Horst G. Zimmer

Journal:
Math. Comp. **48** (1987), 351-370

MSC:
Primary 14G25; Secondary 11D25, 11Y50, 14K15

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866121-6

MathSciNet review:
866121

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using Néron's reduction theory and a method of Tate, we develop a procedure for calculating the local and global Néron-Tate height on an elliptic curve over the rationals. The procedure is illustrated by means of two examples of Silverman and is then applied to calculate the global Néron-Tate height of a series of rank-one curves of Bremner-Cassels and of a series of rank-two curves of Selmer. In the latter case, the regulator is also computed, and a conjecture of S. Lang is investigated numerically. In dealing with the arithmetic of elliptic curves *E* over a global field *K*, the task arises of computing the Néron-Tate height on the group of rational points of *E* over *K*. Solving this task in an efficient manner is important, for instance, in view of calculations concerning the Birch and Swinnerton-Dyer conjecture (see [2]) or of the conjectures of Serge Lang [6]. The purpose of this note is to suggest a procedure for performing the necessary calculations.

**[1]**A. Bremner and J. W. S. Cassels,*On the equation 𝑌²=𝑋(𝑋²+𝑝)*, Math. Comp.**42**(1984), no. 165, 257–264. MR**726003**, https://doi.org/10.1090/S0025-5718-1984-0726003-4**[2]**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**, https://doi.org/10.1090/S0025-5718-1985-0777279-X**[3]**J. W. S. Cassels,*Diophantine equations with special reference to elliptic curves*, J. London Math. Soc.**41**(1966), 193–291. MR**0199150**, https://doi.org/10.1112/jlms/s1-41.1.193**[4]**H. G. Folz,*Ein Beschränktheitssatz für die Torsion von 2-defizienten elliptischen Kurven über algebraischen Zahlkörpern*, Ph.D. Thesis, Saarbrücken, 1985.**[5]**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****[6]**Serge Lang,*Conjectured Diophantine estimates on elliptic curves*, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 155–171. MR**717593****[7]**Serge Lang,*Fundamentals of Diophantine geometry*, Springer-Verlag, New York, 1983. MR**715605****[8]**Ernst S. Selmer,*The Diophantine equation 𝑎𝑥³+𝑏𝑦³+𝑐𝑧³=0*, Acta Math.**85**(1951), 203–362 (1 plate). MR**0041871**, https://doi.org/10.1007/BF02395746**[9]**Joseph H. Silverman,*Lower bound for the canonical height on elliptic curves*, Duke Math. J.**48**(1981), no. 3, 633–648. MR**630588****[10]**John T. Tate,*The arithmetic of elliptic curves*, Invent. Math.**23**(1974), 179–206. MR**0419359**, https://doi.org/10.1007/BF01389745**[11]**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****[12]**J. T. Tate, Letter to J.-P. Serre, Oct. 1, 1979.**[13]**Horst Günter Zimmer,*On the difference of the Weil height and the Néron-Tate height*, Math. Z.**147**(1976), no. 1, 35–51. MR**0419455**, https://doi.org/10.1007/BF01214273**[14]**Horst G. Zimmer,*Quasifunctions on elliptic curves over local fields*, J. Reine Angew. Math.**307/308**(1979), 221–246. MR**534221**, https://doi.org/10.1515/crll.1979.307-308.221**[15]**Horst G. Zimmer,*Torsion points on elliptic curves over a global field*, Manuscripta Math.**29**(1979), no. 2-4, 119–145. MR**545037**, https://doi.org/10.1007/BF01303623**[16]**B. J. Birch and W. Kuyk (eds.),*Modular functions of one variable. IV*, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, 1975. MR**0376533**

Retrieve articles in *Mathematics of Computation*
with MSC:
14G25,
11D25,
11Y50,
14K15

Retrieve articles in all journals with MSC: 14G25, 11D25, 11Y50, 14K15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866121-6

Article copyright:
© Copyright 1987
American Mathematical Society