Computing heights on elliptic curves

We describe how to compute the canonical height of points orn 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. Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equation (1) y2 +alXy+a3Y = X3 +a2X2 +a4X+a6. The canonical height on E is a quadratic form h: E(K) R. (For the definition and basic properties of h, see [11, VIII, Section 9] or [6, Chapter VI].) The canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves, being used for such diverse purposes as studying values of L-functions [5], numbers of integral points [12], and transcendence theory [9]. It is also important as a computational tool, such as its use in Zagier's algorithm for finding integral points up to large bounds [18]. It is thus of interest to have an efficient method for calculating the canonical height of a point. The usual definition of h as a limit h(P) = limn,0 4-nh(x(2nP)) is not practical for computation. Instead, one uses the fact that the canonical height can be written as a sum of local heights, one term for each distinct absolute value on K: (2) h(P) = E nA, (P). vEMK (For example, if K = Q, then MK can be identified with the set of rational primes together with the usual absolute value on Q. The multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field K.) The local height corresponding to a non-Archimedean absolute value is given by intersection theory in a well-known manner. (See, e.g., [2], [4] or [7, Chapter 11, Section 5].) We will describe a quick way to compute non-Archimedean local heights in Section 5. The local height for an Archimedean absolute value is given by a transcendental function, and so efficient computation is somewhat more difficult. J. Tate [15] ?1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page 339 Received August 20, 1987; revised October 21, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 11G05, 14K07, 11D25. *This work was partially supported by NSF grant #DMS-8612393. ** Current address: Mathematics Department, Brown University, Providence, RI 02912. This content downloaded from 207.46.13.103 on Thu, 20 Oct 2016 04:37:48 UTC All use subject to http://about.jstor.org/terms 340 JOSEPH H. SILVERMAN has given an easily computed power series which works for real absolute values. Precisely, for a given curve E and point P = (x, y), he gives a sequence of easily computed numbers co, cl,... so that