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

MathSciNet review:
866121

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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.

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

Article copyright:
© Copyright 1987
American Mathematical Society