Topics in Arithmetic Geometry: Moduli Varieties, L-functions, Arakelov Theory and Their Interactions and Applications

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This project lies in a subfield of mathematics known as arithmetic geometry. Many of the questions in the project are motivated by the philosophy that information about algebraic equations over the integers 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 moduli varieties, L- functions, Arakelov theory and their interactions. This work will have applications to solution to the Birch and Swinnerton-Dyer conjecture, its generalization by Beilinson and Bloch, as well as related problems. . In particular, research will be done on 1) the arithmetic Gan-Gross-Prasad conjecture; 2) heights of Drinfeld-Heegner cycles with arbitrary length; 3) generalization of the Gross-Zagier formula to higher dimensions.