Arithmetic of Shimura Varieties and Applications

这个项目涉及数论和算术几何的研究。这是一个研究领域，通过密码学和编码理论的某些方面应用于网络安全。Shimura簇是一类特殊的高维曲面，其丰富的几何和算法使其成为现代数学的中心研究对象之一。它们在朗兰兹计划中扮演着重要的角色，该计划在理论物理中越来越受到关注，并且是理解椭圆曲线和阿贝尔簇的重要工具，这些曲线和阿贝尔簇现在密码学中发挥着重要作用。首席调查员对Shimura簇的研究的主要目标是证明Faltings Height的新公式，Faltings Height是一种度量椭圆曲线和交换簇复杂性的方法。首席调查员将研究酉Shimura簇和正交Shimura簇的积分模型上的特殊循环的算法。该项目有四个截然不同但又相互关联的部分。第一部分是将Borcherds关于么正Shimura簇上的一系列生成因子的模性的一个定理推广到积分模型。这一推广的主要应用将是证明Cm阿贝尔变种的Faltings高度与Artin L函数的导数之间Colmez猜想关系的新情形。第二部分类似于第一部分，但关于正交的下村品种。这将产生更多关于科尔梅兹猜想的案例。第三部分是将Gross-Kohnen-Zagier定理推广到酉系和正交系的Shimura簇，期望得到Beilison-Bloch猜想的新情形，该猜想将Chow群的维度与L-函数的零阶联系起来。第四个方案涉及到正交Shimura簇的mod_p约化简的显式刻划，这是第三个方案所必需的。

This project concerns investigations in the theory of numbers and arithmetic geometry. This is an area of research that has applications to cyber security, through cryptography, and to some aspects of coding theory. Shimura varieties are particular kinds of higher-dimensional surfaces, and their rich geometry and arithmetic puts them among the central objects of study in modern mathematics. They play an essential role in the Langlands program, which is of increasing interest in theoretical physics, and are important tools for understanding elliptic curves and abelian varieties, which now play a major role in cryptography. The primary goal of the Principal Investigator's research on Shimura varieties is to prove new formulas for Faltings heights, which are a way of measuring the complexity of elliptic curves and abelian varieties.The Principal Investigator will study the arithmetic of special cycles on integral models of unitary and orthogonal Shimura varieties. There are four distinct, but interrelated, parts of the project. The first part is to extend to integral models a theorem of Borcherds on the modularity of a generating series of divisors on an unitary Shimura variety. The main application of this extension will be to prove new cases of Colmez's conjectural relation between Faltings heights of CM abelian varieties and derivatives of Artin L-functions. The second part is similar to the first, but on orthogonal Shimura varieties. This will yield still more cases of Colmez's conjecture. The third part of the project is a generalization of the Gross-Kohnen-Zagier theorem to unitary and orthogonal Shimura varieties, which is expected to yield new cases of the Beilison-Bloch conjecture relating dimensions of Chow groups to orders of vanishing of L-functions. The fourth project involves the explicit description of mod p reductions of orthogonal Shimura varieties, and is necessary for the third project.