Rational Points on Algebraic Varieties

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Wang 9700781 This award funds research into the following two distinct projects. (Project 1) In 1974, S. Lang made a conjecture which connects the geometry (hyperbolicity) of an algebraic variety defined over a number field with the arithmetic (Mordellicity) of the variety. The conjecture is true for curves and subvarieties of Abelian varieties. Little is known for other varieties. Professors Sarnak and Wang have shown that some hypersurfaces yield either a violation of the Hasse principle which is not accounted for by the Brauer--Manin obstruction or a violation of the above Lang's conjecture. Professor Wang will continue to investigate this problem for 0-cycles of degree 1. (Project 2) The densities of rational points on algebraic varieties have been studied extensively. Professor Wang plans to study weak approximation, Brauer--Manin obstruction and the modified Mazur's conjecture on the topology of rational points on the Hurwitz families, and more generally, on Hurwitz spaces. Since every variety is uniformized by Hurwitz spaces, this research will shed light on these problems in the general case. This research falls into the general mathematical field of Number Theory. It concerns the solutions to algebraic equations in whole numbers and generalizations. Number theory is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century, it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.