Arithmetic Intersection on Shimura Varieties and Properties of Abelian Varieties

志村品种的算术交集及阿贝尔品种的性质

基本信息

批准号：
1801237
负责人：
Yunqing Tang
金额：
$ 16.7万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801237&HistoricalAwards=false
关键词：
Arithmetic Intersection Shimura Varieties Properties

项目摘要

This research project concerns work in arithmetic geometry, a branch of mathematics that studies polynomial equations over the integers. Such equations define geometric objects; among them the most accessible and fundamental ones are elliptic curves (one-dimensional case) and abelian varieties (higher-dimensional analogue). A modern way to approach these objects is to study them in families, and certain geometric objects, so-called Shimura varieties, have been used to parametrize these families. Arithmetic properties of abelian varieties can be translated into arithmetic-geometric properties of the corresponding Shimura variety. In this way, the principal investigator and her collaborators will be able to use methods from different areas of mathematics such as algebraic geometry, number theory, and representation theory to study the arithmetic of abelian varieties.The main theme of this project is the infinitude of certain thin sets of primes arising from reduction types of abelian varieties. When the abelian varieties are over number fields, the principal investigator and her collaborators are aiming for results along the line of Elkies' theorem on supersingular reductions of elliptic curves. The research will focus on nonsimple reductions or reductions with higher Picard rank and establish a general framework to treat certain abelian varieties of arbitrarily high dimension. In the function field case, certain new phenomena appear and are related to the geometry of the Newton strata of the corresponding Shimura variety. The philosophy of this project is related to the Kudla program on arithmetic intersection of special cycles on Shimura varieties. In addition to the usual setting of the Kudla program, certain non-special cycles are studied in the framework of this project.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.

该研究项目涉及算术几何的工作，算术几何是研究整数多项式方程的数学分支。这些方程定义了几何对象；其中最容易理解和最基础的是椭圆曲线（一维情况）和阿贝尔簇（高维模拟）。处理这些对象的一种现代方法是在族中研究它们，并且某些几何对象，即所谓的 Shimura 品种，已被用来参数化这些族。阿贝尔簇的算术性质可以转化为相应志村簇的算术几何性质。这样，首席研究员和她的合作者将能够使用代数几何、数论和表示论等不同数学领域的方法来研究阿贝尔簇的算术。这个项目的主题是无穷大由阿贝尔簇的约简类型产生的某些薄素数集。当阿贝尔簇超过数域时，首席研究员和她的合作者的目标是沿着椭圆曲线超奇异约简的 Elkies 定理得到结果。该研究将集中于非简单约简或具有更高皮卡德等级的约简，并建立一个通用框架来处理某些任意高维的阿贝尔簇。在函数场的情况下，出现了某些新现象，并且与相应的Shimura簇的牛顿层的几何形状有关。该项目的理念与 Shimura 品种上特殊循环算术交集的 Kudla 程序相关。除了 Kudla 计划的通常设置之外，该项目的框架中还对某些非特殊周期进行了研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。