Computation of the Néron-Tate height on elliptic curves

Tschöpe, Heinz M.; Zimmer, Horst G.

doi:10.1090/S0025-5718-1987-0866121-6

Computation of the Néron-Tate height on elliptic curves
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by Heinz M. Tschöpe and Horst G. Zimmer PDF

Math. Comp. 48 (1987), 351-370 Request permission

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 $E(K)$ 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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