Mathematical Sciences: The Method of Coleman and Chabauty

麦卡勒姆9624219 科尔曼和Chabauty的方法是一种方法，用于界定曲线上的合理点。在过去的几年里，研究人员通过以精确的方式引入下降来改进这种方法。到目前为止，他对这种方法的研究仅限于费马曲线。他现在将考虑更普遍的应用这种方法，并将开发结果的独立利益在多变量岩泽理论和岩泽理论的p进表示。 广义地说，对科尔曼和夏博蒂方法的改进包括两部分。第一个是一个元素的某一塞尔默组连接到雅可比曲线的建设，第二个是通过一种明确的互反法到一个p-adic解析函数，消失的曲线上的合理的点，这个元素的转换。第二部分，在费马曲线的情况下，为费马大定理（不仅仅是第二种情况）提供了一个新的、漂亮的几何论证，证明了那些素数p存在必要的塞尔默群元素。研究者已经证明了当p是正则的时，存在必要元素。努力构建它更普遍地导致了一些有趣的结果，在多变量岩泽理论。调查员将继续从事这方面的调查工作，既考虑到其独立利益，也考虑到其对一般方法的可能应用。 这个项目福尔斯属于算术代数几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。 事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。在它的许多后果是新的纠错码。 这种代码对于现代计算机（硬盘）和光盘都是必不可少的。

McCallum 9624219 The method of Coleman and Chabauty is a method for bounding rational points on curves. Over the last few years, the investigator has developed a refinement of this method by bringing in descent in a precise way. Until now, his work on this method has been restricted to Fermat curves. He will now consider more general applications of this method, and will develop results of independent interest in multivariable Iwasawa theory and in the Iwasawa theory of p-adic representations. Broadly, the refinement of the method of Coleman and Chabauty consists of two parts. The first is the construction of an element of a certain Selmer group attached to the Jacobian of the curve, and the second is the transformation of this element via a sort of explicit reciprocity law into a p-adic analytic function that vanishes on the rational points of the curve. The second part has, in the case of Fermat curves, yielded a new and beautifully geometric argument for Fermat's Last Theorem (not just the second case) for those primes p for which the necessary Selmer group element exists. The investigator has already proved that the necessary element exists when p is regular. The effort to construct it more generally has led to some interesting results in multi-variable Iwasawa theory. The investigator will continue to work on this line of inquiry, both for its independent interest and for its potential application to the general method. This project falls into the general area of Arithmetic Algebraic Geometry-a subject that blends two of the oldest areas of mathematics: Number Theory and Geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.