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。志村簇是高维曲面的特殊类型，它们丰富的几何和算术使它们成为现代数学研究的中心对象。 它们在朗兰兹计划中发挥着至关重要的作用，朗兰兹计划在理论物理学中越来越受到关注，并且是理解椭圆曲线和阿贝尔簇的重要工具，现在在密码学中发挥着重要作用。 主要研究员对志村簇的研究的主要目标是证明新的Faltings高度公式，这是一种测量椭圆曲线和阿贝尔簇复杂性的方法。主要研究员将研究酉和正交志村簇积分模型上的特殊圈的算法。 该项目有四个不同但相互关联的部分。 第一部分是将Borcherds关于酉Shimura簇上因子生成列的模性定理推广到积分模型。 这个扩展的主要应用将是证明新的情况下Colmez的Faltings高度CM交换品种和Artin L-函数的衍生物之间的关系。 第二部分与第一部分相似，但在正交Shimura品种上。 这将产生更多的情况下科尔梅兹猜想。 该项目的第三部分是将Gross-Kohnen-Zagier定理推广到酉和正交Shimura簇，这有望产生Beilison-Bloch猜想的新情况，该猜想将Chow群的维数与L-函数的消失阶联系起来。 第四个项目涉及到明确的描述模p减少正交志村品种，是必要的第三个项目。

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.