On the method of Coleman and Chabauty
On the method of Coleman and Chabauty
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论科尔曼和查博蒂的方法
DOI：
10.1007/bf01459799
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发表时间：
1994
影响因子：
1.4
通讯作者：
William G. McCallum
中科院分区：
文献类型：
--
作者：
William G. McCallum
设C是定义在数域K上的亏格g>；2的曲线，J是C的Jacobian，Coleman[C2]在Chabauty的基础上证明了当Mordell-Weil群J(K)的秩r小于g时，如何得到C(K)的基数的良界.该方法的关键是对K的某个赋值v构造一个对数，其核包含J(K)，其对C(Kv)的限制显式地表示为微分的积分.本文试图通过对费马曲线的详细研究，证明这种方法可以成为确定曲线上有理点的一种非常精确的工具。我们证明了如何将19次Fermat曲线的Jacobian的Selmer群的一个元素转化为该曲线本身上的p-进解析函数，其零点集包含所有有理点。作为结果，我们证明了正则素数的费马最后定理的第二种情况，该方法依赖于Selmer群中是否存在合适的元素；由于缺乏令人满意的关于Fermat曲线的雅可比下降的理论，我们只能证明当p是正则的情况下该元素存在。当然，在这种情况下，库默已经证明了费马大定理的全部。然而，我们相信这篇论文的兴趣在于方法，而不是定理，因此独立于库曼纳，也独立于Wiles最近的工作。我们的方法是不同的，它提供了Coleman和Chabauty方法的发展，其中许多方面可以推广到任意曲线，尽管我们不试图在这里进行这种推广。我们现在更详细地描述这篇论文的内容。设p是奇素数，F是Pm-Fermat曲线，具有射影方程
Let C be a curve of genus g >_ 2, defined over a number field K , and let J be the Jacobian of C. Coleman [C2], following Chabauty, has shown how to obtain good bounds on the cardinality of C(K) if the rank r of the Mordell-Weil group J(K) is less than g. The key to the method is to construct a logarithm on J(Kv), for some valuation v of K , whose kernel contains J(K), and whose restriction to C(Kv) is represented explicitly as the integral of a differential. This paper is an attempt to make the case, through a detailed examination of the case of Fermat curves, that this method can be fashioned into a quite precise tool for bounding rational points on curves. We show how to transform an element of the Selmer group of the Jacobian of a Fermat curve of degree 19 into a p-adic analytic function on the curve itself, whose zero set contains all the rational points. As a consequence, we prove the second case of Fermat's Last Theorem for regular primes, The method depends on the existence of a suitable element in the Selmer group; for the lack of a satisfactory theory of descent for Jacobians of Fermat curves, we can only show that this element exists in the case that p is regular. Of course, in that case, Kummer had already proved the whole of Fermat's Last Theorem. However, we believe the interest of this paper is in the method, not the theorem, and as such is independent of Kumaner, and also of the recent work of Wiles. Our method is different, and offers a development of the method of Coleman and Chabauty, many aspects of which are generalizable to arbitrary curves, although we do not attempt to make that generalization here. We now describe the contents of the paper in more detail. Let p be an odd prime, and let F be the pm Fermat curve, with projective equation