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在不同类型的LDS，例如数字字段，有限字段，全局功能字段和扩展方面定义的几何对象的算术特性，例如曲线和Abelian品种。调查员及其合作者将把物体带入各种类型的家庭，而空间的几何形状将原始对象的theduce属性参数化。这些特殊的物体和这些特殊的物体和重要的特性使它们在许多领域的研究中的主要主题，而算术几何学的一些目标将推广到重要的数学家工作。 E学生研究校长将监督的研究是主要研究者及其合作者在项目中追求的两个主要方向功能领域。在基本场所的全球领域定义的研究项目中，代数家族品种的位置是由光滑的全球田地范围的参数。在某些具有非平凡性内态群体的Abelian品种中，普通素数的集合具有密度1。在相反的方向上，另一个项目旨在限制Thelian品种接受基本还原曲线的许多素数。该奖项反映了NSF的任务，并通过使用Toundation的知识分子优点和更广泛的影响审查标准来评估，这反映了NSF的任务，这反映了NSF的任务值得支持。