Arithmetic on Shimura Varieties and Applications

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This research 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. In more detail, the investigator will study the connections between arithmetic on a Shimura variety and L-functions and Eisenstein series. In different projects, he aims to prove various arithmetic Siegel-Weil formulas, determine when a special divisor is cohomologically trivial but not rational trivial in a Picard modular surface, prove a conjecture of Gross and Zagier on algebraicity of CM values, continue to work on Colmez conjecture, and study the rationality and integrality of unitary modular forms in terms of their Fourier-Jacobi coefficients. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.