Arithmetic Intersection on Shimura Varieties and Properties of Abelian Varieties

这项研究项目涉及算术几何方面的工作，算术几何是研究整数上的多项式方程的数学分支。这样的方程定义了几何对象；其中最容易接近和最基本的是椭圆曲线(一维情况)和阿贝尔簇(高维类比)。一种处理这些对象的现代方法是在族中研究它们，而某些几何对象，即所谓的下村簇，已经被用来对这些族进行参数化。交换簇的算术性质可以转化为相应的Shimura簇的算术几何性质。这样，主要研究人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变种的算术。这个项目的主要主题是由阿贝尔变种的约化类型产生的某些稀疏素数集的无穷大。当阿贝尔变种在数域上时，主要研究人员和她的合作者的目标是沿着Elkies关于椭圆曲线的超奇异约化定理的方向得出结果。这项研究将集中于非简单的约化或具有更高Picard秩的约化，并建立一个一般框架来处理某些任意高维的交换变种。在函数域的情况下，某些新的现象出现，并与相应的下村变种的牛顿地层的几何有关。这个项目的理念是关于下村品种特殊循环的算术交集的Kudla程序。除了库德拉项目的常规设置外，还在该项目的框架内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智力优势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.