The main goal of this research is to study a conjecture of Mazur concerning the topology of rational points on a variety over the rational number field. Questions related to the Hasse Principle, weak approximation and the Brauer-Manin obstruction play a key role in this study. The approach depends on a detailed study of special properties of each variety under consideration and the construction of canonical height functions when applicable. This research is in the general mathematical area of Arithmetic Geometry. This ultramodern research area combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.
