The Frobenius action on curves and abelian varieties

曲线和阿贝尔簇上的弗罗贝尼乌斯作用

• 批准号: 2302511
• 负责人: Wanlin Li
• 金额: $ 18.89万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302511&HistoricalAwards=false
• 关键词: Frobenius; action; curves; abelian; varieties

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.

该研究项目旨在研究几何对象的算术属性，例如在数域、有限域、全局函数域及其扩展等不同类型域上定义的曲线和阿贝尔簇。首席研究员和她的合作者将把这些对象打包成各种类型的族，然后使用参数化这些族的空间几何来推断原始对象的属性，这是首席研究员和她的合作者想要回答的主要问题。是估计数字这些特殊对象的出现频率以及它们出现的频率或罕见程度，这些特殊对象呈现出有用且重要的属性，使它们成为数论和算术几何许多领域和方向研究的中心主题，一些目标结果将概括重要的先前工作。该研究计划将提供许多适合本科生和研究生研究的项目，由首席研究员负责监督。首席研究员和她的合作者将在这些项目中追求两个主要方向，即研究 p 整除数。高年级家庭团体维阿贝尔簇并研究全局函数域某些族的理想类群的结构。研究项目中有不同类型的族，例如在由位置参数化的全局域上定义的阿贝尔簇的约简。具体而言，一个项目旨在证明某些阿贝尔簇的约简中的普通素数集。非平凡自同态群的密度为 1。在相反的方向上，另一个项目旨在构造无限多个素数，在这些素数处这些阿贝尔簇允许基本约简，推广 Elkies 关于椭圆曲线的无穷多个超奇异素数的工作 对于理想类群，首席研究员和她的合作者将使用伽罗瓦上同调和计算工具来预测和证明有理函数域的 l 次扩展的 l 扭转类分布的属性。该奖项通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。