Zero-Cycles over Arithmetic Fields and Reciprocity Laws

几乎每一门科学的核心问题都是对具有相似特征的物体进行分类。在代数几何中，数学家对代数簇的分类感兴趣，即多项式方程的解的集合。这种分类是通过确定和计算品种的某些不变量来进行的。这样的不变量可以是数值的，或者更常见的是具有特定代数或几何结构的集合。更几何的方法集中在复数上的代数簇，它们本身具有丰富的几何。另一方面，在数论中，数学家对寻找多项式方程的整数或有理解感兴趣;这一领域的一个著名例子是费马最后定理。在复数集合中，有理数的集合非常稀少，这就是为什么这样的解很难被检测到。这个项目的目的是研究一个几何不变量，称为周群的零圈，既涉及到分类问题，也涉及到算术的合理解决方案。该项目的主要目标是研究当我们研究有理数或真实的数的算术模拟（即p-adic数）时，处理这个群的结构的结构。在这个项目中的方法将涉及算术和代数几何以及K理论的技术。该项目的重点是阿贝尔变种的研究，阿贝尔变种是一类具有一些额外结构的变种。该项目涉及的“局部问题”的品种超过p-adic数，其中使用p-adic霍奇理论将是关键。当地计划的主要目标之一是建立Colliot-Thélène猜想。其次，该项目还涉及有理数上的各种“全局问题”。全球计划的目标将是证明贝林森的一个猜想。最后，该项目将调查一个与障碍问题有关的“地方到全球”方案。其目的是证明加藤和齐藤的一个猜想，该猜想有可能导致新型欧拉系统的构建。该奖项反映了NSF的法定使命，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。

A central question in almost every science is the classification of objects that feature similar characteristics. In algebraic geometry, mathematicians are interested in the classification of algebraic varieties, that is, sets of solutions to polynomial equations. This classification is carried out by determining and computing certain invariants of the variety. Such invariants can be numerical, or more often, sets that have a specific algebraic or geometric structure. The more geometric approach focuses on algebraic varieties over the complex numbers, which themselves have a rich geometry. In number theory, on the other hand, mathematicians are interested in finding integer or rational solutions to polynomial equations; a famous example in this area is Fermat's last theorem. The set of rational numbers is very scarce within the set of complex numbers, which is what makes such solutions so hard to detect. This project aims to study a geometric invariant, called the Chow group of zero-cycles, that relates both to the classification problem and to the arithmetic of rational solutions. The main goal of the project is to investigate conjectures that deal with the structure of this group when we work over the rational numbers or over the arithmetic analog of the real numbers, namely the p-adic numbers. The methods in this project will involve techniques from arithmetic and algebraic geometry as well as K-theory. The project focuses on the study of abelian varieties, a class of varieties that has some extra structure. The project involves "local questions" for varieties over the p-adic numbers, where the use of p-adic Hodge theory will be the key. Among the main goals of the local program is to establish a conjecture of Colliot-Thélène. Second, the project involves also "global questions" for varieties over the rational numbers. The goal of the global program will be to prove a conjecture of Beilinson. Lastly, the project will investigate a "local-to-global" program related to obstruction questions. The goal is to prove a conjecture of Kato and Saito, which could potentially lead to the construction of a new type of Euler system.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.