Topics in arithmetic geometry

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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.