Topics in Arithmetic Algebraic Geometry

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The investigator and his collaborators study the following topics in arithmetic algebraic geometry:the uniformity of small points for family of heights,the arithmetic Kodaira-Spencer class for arithmetic surfaceswith applications to the effective Mordell conjecture, the Gross-Zagier type formula for Shimura varietieswith applications to the Birch and Swinnerton-Dyer conjecture andthe Andre-Oort conjecture.This is a project in a subfield of mathematics known asnumber theory. Many of these questions are motivated by the philosophy that algebraic information can be obtained by geometric methods. At the center of this projectis the use of a symmetric group, or algebraic group,which is an object that is both algebraic and geometricin nature. These symmetric groups arise naturally in physics and chemistry. It is not too ambitious to say that the solution to the problems in this proposal will one day affect research in cryptography,theoretical physics, and quantum computing.