Bounding the number of rational points on certain curves of high rank

Let K be a number field and let C be a curve of genus g > 1 defined over K. In this dissertation we describe techniques for bounding the number of K-rational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reduction or the existence of a known rational point on the curve. We have also attempted to clearly state the circumstances under which Chabauty techniques can be applied. Our primary goal is to provide a flexible and powerful tool for computing on specific curves. In Chapter II we develop a technique which, given a K-rational isogeny to the Jacobian of C, produces a positive integer n and a collection of covers of C with the property that the set of K-rational points in the collection is in n-to-1 correspondence with the set of K-rational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of K-rational points on C. The examples in Chapters I and II are taken from problem VI.17 in the Arabic text of the Arithmetica. Chapter III is devoted to the background calculations for this problem. When we assemble the pieces, we discover that the solution given by Diophantus is the only positive rational solution to this problem.