The difference between the Weil height and the canonical height on elliptic curves
The difference between the Weil height and the canonical height on elliptic curves
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椭圆曲线上的韦尔高度和规范高度之间的差异
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发表时间：
1990
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通讯作者：
J. Silverman
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作者：
J. Silverman
对椭圆曲线上点的韦尔高度和规范高度之差的估计可用于多种目的，包括理论和计算。在本文中，我们根据椭圆曲线的 j 不变量和判别式给出了这种差异的明确估计。 Serge Lang 提出的证明方法是将规范高度分解为局部高度之和。我们通过三个例子来计算 Mordell-Weil 群的生成元来说明我们的估计的一种用途。设 E 是在数域 K 上定义的椭圆曲线，由 Weierstrass 方程 (1) 给出，其中 A 和 B 在 K 的整数环中。E 上的规范高度是二次形式 h：E(K)-+ R。（有关 h 的定义和基本属性，请参阅 [10，第八章，?9 或 6，第六章]。）规范高度由该属性共同决定事实上，差异
Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the j-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples. Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equation (1) y2 =x +Ax+B with A and B in the ring of integers of K. The canonical height on E is a quadratic form h: E(K)-+ R. (For the definition and basic properties of h, see [10, Chapter VIII, ?9 or 6, Chapter VI].) The canonical height is determined by this property together with the fact that the difference