Mathematical Sciences: The Method of Coleman and Chabauty

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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.