The Frobenius action on curves and abelian varieties

本课题旨在研究在不同类型的域（如数域、有限域、全局函数域及其扩展）上定义的曲线和阿贝尔变量等几何对象的算术性质。首席研究员和她的合作者不是一次研究一个单独的物体，而是将这些物体打包成各种类型的族，然后使用空间的几何参数化这些族来推断原始物体的属性。首席研究员和她的合作者想要回答的主要问题是，估计这些家族中特殊物体的数量，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研究中心课题。一些目标结果将推广其他数学家先前的重要工作。该研究计划将提供许多适合本科生和研究生研究的项目，由首席研究员指导。项目的主要研究方向有两个，即研究高维阿贝尔变族的p可分群和研究全局函数域某些族的理想类群的结构。在研究项目中有不同类型的族，如由基域的位置参数化的全局域上定义的阿贝尔变种的约简，由志村变种参数化的阿贝尔变种的代数族，以及由它们的判别式排序的全局函数域集。具体而言，一个项目旨在证明具有非平凡自同态群的某些阿贝尔变体的约简中的普通素数集具有密度1。在相反的方向上，另一个项目旨在构造无限多个素数，在这些阿贝尔变体上允许基本约简，推广Elkies关于椭圆曲线的超奇异素数的无限的工作。对于理想类群，主要研究者和她的合作者将使用伽罗瓦上同调和计算工具来预测和证明有理函数场的l阶扩展的l-扭转类分布的性质。该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 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.