Arithmetic on Shimura Varieties and Applications

这项研究项目是数学的一个子领域，称为算术几何。该项目中的许多问题都是基于这样一种哲学，即关于整数上的代数方程的信息可以通过几何方法获得。该项目所研究的问题的解决将对密码学、理论物理和量子计算的研究产生实质性的影响。更详细地，研究人员将研究Shimura变种上的算术与L函数和艾森斯坦级数之间的联系。在不同的项目中，他的目标是证明各种算术Siegel-Weil公式，确定Picard模曲面中的特殊因子何时是上同调平凡但不是有理平凡的，证明Gross和Zagier关于CM值的代数性的猜想，继续研究Colmez猜想，并根据其傅立叶-雅可比系数研究酉模形式的合理性和完整性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.