Rational Points and Heights

New York University, New York NY

Investigators

Abstract

The project addresses questions in Arithmetic Algebraic Geometry. It is proposed to study the distribution of rational points on algebraic varieties over number fields, and to explore the interplay between global geometric invariants and arithmetic properties of varieties. The focus is on proving potential density for families of varieties, such as Fano varieties or Calabi-Yau varieties, and on establishing asymptotic formulas for the number of rational points of bounded height on equivariant compactifications of groups and homogeneous spaces. The proposed research ranges from explicit numerical experiments to advanced problems in modern and rapidly developing areas of mathematics. This work will have a broader impact on the education of students and training of young mathematicians, as there is a rich supply of concrete questions and examples, consolidating our understanding of the subject and leading to new structures. Increasingly, applications of Arithmetic Algebraic Geometry are spreading to vital areas of information storage, processing and transmission; these applications rely on further advances on fundamental problems to be addressed in this proposal.