Computing heights on elliptic curves
Computing heights on elliptic curves
复制标题
计算椭圆曲线上的高度
DOI：
10.1090/s0025-5718-1988-0942161-4
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发表时间：
1988
影响因子：
2
通讯作者：
J. Silverman
中科院分区：
文献类型：
--
作者：
J. Silverman
本文介绍如何计算椭圆曲线上点的标准高。泰特给出了R上阿基米德局部高度的一个快速收敛的级数。我们描述了一个修改后的版本泰特的系列也收敛于C，并给出了一个有效的程序计算当地的高度在非阿基米德的地方。通过这种方式，我们可以计算具有复杂嵌入的数域的高度。我们也给出了明确的估计，我们的系列的尾部，并提出了几个例子。设E是定义在数域K上的椭圆曲线，例如由Weierstrass方程（1）y2 + a1 Xy + a3 Y = X3 + a2 X2 + a4 X +a6给出。E上的标准高度是一个二次型h：E（K）R。(For h的定义和基本性质，见[11，VIII，第9节]或[6，第VI章]。标准高度是椭圆曲线算术理论中一个极其重要的理论工具，被用于研究L函数的值[5]，整点的数目[12]和超越理论[9]等不同的目的。它也是一个重要的计算工具，例如它在Zagier算法中的使用，用于寻找大边界的积分点[18]。因此，有一个有效的方法来计算一个点的典范高度是感兴趣的。通常把h定义为极限h（P）= limn，0 4-nh（x（2 π））对于计算是不实际的.相反，我们使用这样一个事实，即正则高度可以写成局部高度的和，K上每个不同的绝对值对应一项：（2）h（P）= E nA，（P）。vEMK（例如，如果K = Q，那么MK可以用有理素数的集合以及Q上的通常绝对值来标识。选择重数nv，以便积公式成立，并且h独立于场K的选择。）对应于非阿基米德绝对值的局部高度由相交理论以众所周知的方式给出。(See例如，在一个实施例中，[2]，[4]或[7，第11章，第5节]。）我们将在第5节描述一种快速计算非阿基米德局部高度的方法。阿基米德绝对值的局部高度由超越函数给出，因此有效的计算有点困难。J. Tate [15]？1988美国数学学会0025-5718/88 $1.00 + $.25 per page 339接收于1987年8月20日;修订于1987年10月21日。1980年数学学科分类（1985年修订）。初级11 G 05、14 K 07、11 D25。* 这项工作得到了NSF资助#DMS-8612393的部分支持。** 目前地址：布朗大学数学系，普罗维登斯，RI 02912。此内容于2016年10月20日星期四04：37：48 UTC从 207.46.13.103下载所有使用http://about.jstor.org/terms约束340 JOSEPH H. SILVERMAN给出了一个易于计算的幂级数，它适用于真实的绝对值。确切地说，对于一个给定的曲线E和点P =（x，y），他给出了一个序列的容易计算的数字co，cl，.使得
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