THE METHOD OF CHABAUTY AND COLEMAN WILLIAM MCCALLUM AND
THE METHOD OF CHABAUTY AND COLEMAN WILLIAM MCCALLUM AND
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Chabauty 和 Coleman William McCallum 的方法
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
B. Poonen
中科院分区：
文献类型：
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作者：
B. Poonen
这是对Chabauty和科尔曼的方法的介绍，这是一种p-adic方法，试图确定亏格g ≥ 2的给定曲线上的有理点集。我们提出的方法，给出了几个例子，它在实践中的实施，并讨论其有效性。附录处理曲线复位不良的情况。1.亏格≥ 2的曲线上的有理点我们将研究有理数域Q，尽管我们所说的一切都可以适当地推广到数域。设Q是Q的一个代数闭包。对于每个有限素数p，设Qp是p-adic数域（定义见[Kob 84]）。曲线将被假定为光滑的、射影的和几何积分的。设X是亏格为g ≥ 2的曲线.我们假设X表示为齐次多项式显式有限集的某个P中的零点集。我们可以给出A中一条奇异（但仍是几何积分的）曲线的方程;在这种情况下，可以理解X是这条奇异曲线的光滑射影曲线。X上的有理点可以通过给出它们的坐标来指定。（如果使用X的奇异模型，则可能需要更多的数据。）设X（Q）是X上的有理点集。1922年L. Mordell [Mor 22]证明了X（Q）是有限的，1983年G. Faltings [Fal83].因此，我们有以下定义明确的问题：给定如上所述亏格≥ 2的X，计算X（Q）。A的一个论点。N. Parshin（见[Szp 85]）表明，法尔明斯的证明可以适用于给出X（Q）的基数上界。但是，在这个意义上说，Faltings的证明仍然是无效的，因为它没有提供一个算法来找到X（Q）中的点，甚至在原则上。事实上，目前还不知道是否有任何算法可以保证解决这个问题。即使g = 2的情况也很难。然而，有一些技术可以应用：参见[Poo 02]的调查。在个别曲线上，这些似乎经常解决问题，甚至在使用时总是解决问题日期：2010年6月14日。2000年数学学科分类。小学11 G30;中学14 G 05，14 K20。
This is an introduction to the method of Chabauty and Coleman, a p-adic method that attempts to determine the set of rational points on a given curve of genus g ≥ 2. We present the method, give a few examples of its implementation in practice, and discuss its effectiveness. An appendix treats the case in which the curve has bad reduction. 1. Rational points on curves of genus ≥ 2 We will work over the field Q of rational numbers, although everything we say admits an appropriate generalization to a number field. Let Q be an algebraic closure of Q. For each finite prime p, let Qp be the field of p-adic numbers (see [Kob84] for the definition). Curves will be assumed to be smooth, projective, and geometrically integral. Let X be a curve over Q of genus g ≥ 2. We suppose that X is presented as the zero set in some P of an explicit finite set of homogeneous polynomials. We may give instead an equation for a singular (but still geometrically integral) curve in A; in this case, it is understood that X is the smooth projective curve birational to this singular curve. Rational points on X can be specified by giving their coordinates. (A little more data may be required if a singular model for X is used.) Let X(Q) be the set of rational points on X. In 1922 L. Mordell [Mor22] conjectured that X(Q) is finite, and in 1983 this was proved by G. Faltings [Fal83]. Thus we have the following well-defined problem: Given X of genus ≥ 2 presented as above, compute X(Q). An argument of A. N. Parshin (see [Szp85]) shows that Faltings’ proof can be adapted to give an upper bound on the cardinality of X(Q). But Faltings’ proof is still ineffective in the sense that it does not provide an algorithm for finding the points in X(Q), even in principle. In fact, it is not known whether any algorithm is guaranteed to solve this problem. Even the case g = 2 seems hard. Nevertheless there are a few techniques that can be applied: see [Poo02] for a survey. On individual curves these seem to solve the problem often, perhaps even always when used Date: June 14, 2010. 2000 Mathematics Subject Classification. Primary 11G30; Secondary 14G05, 14K20.