Topics in arithmetic geometry

这是一个被称为算术几何的数学子领域的项目。该项目中的许多问题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。本课题所研究的问题的解决将对密码学、理论物理和量子计算等领域的研究产生重大影响。研究人员将研究Shimura变种上特殊循环的算术交数及其与L级数自同构的特定值的关系。这项工作将应用于关于交换簇的Birch和Swinnerton-Dyer猜想和关于代数圈的Beilinson-Bloch猜想。

This is a project in a subfield of mathematics known as arithmetic geometry. Many of the questions in the project are motivated by the philosophy that algebraic information can be obtained by geometric methods. Solution to the problems under study in this project will have substantial impact on research in cryptography, theoretical physics, and quantum computing.The investigator will study arithmetical intersection numbers of special cycle on Shimura varieties and their relations to the special values of automorphic of L-series. This work will have applications to the Birch and Swinnerton-Dyer conjecture on abelian varieties and the Beilinson--Bloch conjecture on algebraic cycles.