Topics in Arithmetic Algebraic Geometry

研究者和他的合作者研究了算术代数几何中的以下主题：高度族的小点的均匀性，算术曲面的算术Kodaira-Spencer类及其在有效Mordell猜想中的应用，Shimura变分的Gross-Zagier型公式及其在Birch和Swinnerton-Dyer猜想和Andre-Oort猜想中的应用。这是数学子领域数论的一个项目。许多这样的问题是由这样一种哲学所激发的，即代数信息可以通过几何方法获得。在这个项目的中心使用对称群，或代数群，这是一个对象，代数和几何性质。这些对称群在物理和化学中自然产生。可以毫不夸张地说，本提案中问题的解决方案有一天会影响密码学、理论物理和量子计算的研究。

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.