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。志村变体是一类特殊的高维曲面，其丰富的几何和算术特征使其成为现代数学研究的中心对象之一。它们在朗兰兹程序中起着至关重要的作用，朗兰兹程序在理论物理学中日益引起人们的兴趣，它们是理解椭圆曲线和阿贝尔变分的重要工具，而椭圆曲线和阿贝尔变分现在在密码学中起着重要作用。志村变数研究的主要目标是证明法尔廷高度的新公式，这是测量椭圆曲线和阿贝尔变数复杂性的一种方法。首席研究员将研究酉型和正交Shimura型积分模型上的特殊循环算法。该项目有四个不同但相互关联的部分。第一部分将Borcherds关于酉Shimura变数上生成的因子级数的模性的定理推广到积分模型。这一推广的主要应用将是证明CM阿贝尔变体的Faltings高度与Artin l -函数的导数之间的Colmez猜想关系的新情况。第二部分类似于第一部分，但对志村正交品种。这将产生更多的证明科尔梅兹猜想的例子。该项目的第三部分是将Gross-Kohnen-Zagier定理推广到酉型和正交Shimura变型，这有望产生有关Chow群的维数与l -函数消失阶数的Beilison-Bloch猜想的新情况。第四个项目涉及对正交志村品种的mod p约简的明确描述，这是第三个项目所必需的。

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.