Topics in Arithmetic Algebraic Geometry

这位研究者和他的合作者研究了算术代数几何中的下列主题：高度族的小点的一致性，算术曲面的算术Kodaira-Spencer类及其对有效的Mordell猜想的应用，Shimura变种的Gross-Zagier型公式及其对Birch和Swinnerton-Dyer猜想和Andre-Oort猜想的应用。这是数论数学的一个子领域的项目。这些问题中的许多都是基于这样一种哲学，即代数信息可以通过几何方法获得。在这个投影的中心是使用对称群或代数群，这是一个既具有代数性质又具有几何性质的对象。这些对称基团在物理和化学中自然产生。可以说，这项提案中问题的解决有朝一日会影响密码学、理论物理和量子计算的研究，这一点并不过分雄心勃勃。

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.