Arithmetic Intersection on Shimura Varieties and Properties of Abelian Varieties

这个研究计画是关于算术几何的工作，算术几何是数学的一个分支，研究整数上的多项式方程式。这些方程定义了几何对象;其中最容易理解和最基本的是椭圆曲线（一维情况）和阿贝尔变种（高维模拟）。一个现代的方法来处理这些对象是研究他们的家庭，和某些几何对象，所谓的志村品种，已被用来参数化这些家庭。阿贝尔簇的算术性质可以转化为相应的志村簇的算术几何性质。通过这种方式，主要研究者和她的合作者将能够使用不同数学领域的方法，如代数几何，数论和表示论来研究阿贝尔簇的算术。这个项目的主题是由阿贝尔簇的约化类型产生的某些瘦素数集的无限性。当阿贝尔品种超过数域，主要研究者和她的合作者的目标是结果沿着线埃尔基定理的超奇异减少椭圆曲线。本文的研究将集中在非单约化和具有较高Picard秩的约化上，并建立一个一般的框架来处理任意高维的阿贝尔簇。在函数场的情况下，某些新的现象出现，并与相应的志村品种的牛顿地层的几何形状。这个项目的理念是与Kudla程序的算术相交的特殊周期志村品种。除了Kudla计划的通常设置之外，在该项目的框架内还研究了某些非特殊周期。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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 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.