Height difference bounds for elliptic curves over number fields
Height difference bounds for elliptic curves over number fields
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DOI：
10.1016/j.jnt.2005.03.001
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发表时间：
2006-01-01
影响因子：
0.7
通讯作者：
Siksek, S
中科院分区：
文献类型：
--
作者：
Cremona, JE;Prickett, M;Siksek, S
设E是数域K上的椭圆曲线。设h是E上的对数（或Weil）高度，（h）over cap是E上的正则高度。在帽上差h -（h）的界具有重要的理论和实际意义。可以将h-L分解为K的上半部分上的连续有界函数Psi（上半部分）：E（K-上半部分）-> R的加权和。一个标准的方法来界定h -（h）在上限，（由于朗，和以前采用的西尔弗曼）是绑定每个函数Psi（upper）和总和这些当地的“贡献”。本文给出了非阿基米德上挠时Psi（上挠）极值的简单计算公式，并利用上挠曲线的Tamagawa指数和科代拉符号给出了Psi（上挠）极值的计算公式。对于真实的阿基米德v，Siksek [Rocky Mountain J. Math.25（4）（1990）1501]以前给出了一种严格限制Psi（upper）的方法。我们补充这一点，给出了两种方法，严格限制Psi（upperdance）复杂的阿基米德upperdance。(c)2005年爱思唯尔公司All rights reserved.
Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and (h) over cap be the canonical height on E. Bounds for the difference h - (h) over cap are of tremendous theoretical and practical importance. It is possible to decompose h - L as a weighted sum of continuous bounded functions Psi(upsilon) : E(K-upsilon) -> R over the set of places upsilon of K. A standard method for bounding h - (h) over cap, (due to Lang, and previously employed by Silverman) is to bound each function Psi(upsilon) and sum these local 'contributions'. In this paper, we give simple formulae for the extreme values of Psi(upsilon), for non-archimedean upsilon in terms of the Tamagawa index and Kodaira symbol of the curve at upsilon. For real archimedean v a method for sharply bounding Psi(upsilon) was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Psi(upsilon) for complex archimedean upsilon. (c) 2005 Elsevier Inc. All rights reserved.