Distribution of Rational Points: A Survey

One of the most important areas of investigation in number theory is the study of the distribution of rational or integral points on algebraic varieties. It is not a small historical miracle that an area of investigation in mathematics that started some 2000 years ago would still be of great interest. This field now has profound connections to various areasof modern mathematics including algebraic geometry, complex analysis, logic, and more recently harmonic analysis and automorphic representation theory. One of the central themes in the theory is to explore the relationship between geometric and arithmetic properties of algebraic varieties. One of the guiding principles is the idea that the roughgeometric classification of algebraic varieties according to the ampleness of the canonical (respectively anticanonical) line bundle should directly influence the arithmetic properties.