Topics in Arithmetic Algebraic Geometry

研究者和他的合作者研究算术代数几何中的以下主题：高度族的小点的一致性，算术表面的算术Kodaira-Spencer类及其对有效Mordell猜想的应用，志村变量的Gross-Zagier型公式及其对Birch和Swinnerton-Dyer猜想和Andre-Oort猜想的应用。许多这些问题的动机是哲学，代数信息可以获得几何方法。在这个项目的中心是使用一个对称群，或代数群，这是一个对象，这是代数和几何性质。这些对称群在物理学和化学中自然出现。可以毫不夸张地说，这个提案中的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.