The Frobenius action on curves and abelian varieties

本研究项目旨在研究定义在不同类型的域(如数域、有限域、全局函数域及其扩张)上的几何对象(如曲线和阿贝尔簇)的算术性质。首席研究员和她的合作者不是一次一个地研究每个单独的对象，而是将这些对象打包到各种类型的族中，然后使用将这些族参数化的空间几何来推断原始对象的属性。首席研究员和她的合作者打算回答的主要问题是估计这些家庭中特殊物体的数量以及它们出现的频率或罕见程度。这些特殊的物体呈现出有用和重要的性质，使它们成为数论和算术几何许多领域和方向的中心研究主题。一些目标结果将概括其他数学家以前的重要工作。研究计划将提供许多适合本科生和研究生研究的项目，首席研究员将指导这些项目。主要研究人员和她的合作者将在项目中追求两个主要方向，即研究高维阿贝尔变种家庭的p-可除群和研究某些全局函数域族的理想类群的结构。在研究项目中有不同类型的族，例如由基场的位置参数化为全局域上的阿贝尔簇的约化，由Shimura簇参数化的阿贝尔簇的代数族，以及按其判别式排序的全局函数域的集合。具体地说，一个项目的目的是证明某些具有非平凡自同态群的交换簇的约化中的普通素数集具有密度1。相反，另一个项目的目的是构造无穷多个素数，使这些交换簇的基约化，推广了Elkies关于椭圆曲线的超奇异素数无穷多的工作。对于理想类群，首席研究员和她的合作者将使用伽罗华上同调和计算工具来预测和证明有理函数域的L次扩张的L-扭类的分布性质。这一奖项反映了美国国家科学基金会的法定使命，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 aims to study arithmetic properties of geometric objects such as curves and abelian varieties defined over different types of fields such as number fields, finite fields, global function fields and their extensions. Instead of studying each individual object one at a time, the principal investigator and her collaborators will take these objects and pack them into various types of families, and then use the geometry of the spaces parameterizing these families to deduce properties of the original objects. The main question that the principal investigator and her collaborators aim to answer is to estimate the number of special objects in these families and how often or rarely they occur. These special objects present useful and important properties making them central topics of research in many areas and directions in number theory and arithmetic geometry. Some of the target results will generalize important prior work of other mathematicians. The research program will provide many projects suitable for undergraduate and graduate students research which the principal investigator will supervise.There are two main directions the principal investigator and her collaborators will pursue with the projects, namely, to study the p-divisible groups for families of high dimensional abelian varieties and to study the structure of the ideal class groups of certain families of global function fields. There are different types of families in the research projects, such as the reductions of an abelian variety defined over a global field parameterized by the places of the base field, algebraic families of abelian varieties parameterized by a Shimura variety and sets of global function fields ordered by their discriminant. Specifically, one project aims to prove the set of ordinary primes in the reduction of certain abelian varieties with nontrivial endomorphism groups has density 1. In the opposite direction, another project aims to construct infinitely many primes at which these abelian varieties admit basic reduction, generalizing the work of Elkies’ on the infinitude of supersingular primes for elliptic curves. For ideal class group, the principal investigator and her collaborators will use Galois cohomology and computational tools to predict and prove properties of the distribution of l-torsion classes for degree l extensions of the rational function field.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.