Zero-cycles over local and global fields

数论中的一个经典问题是一个给定的有理数系数多项式方程组是否有一个有理数解。多项式方程的解集产生了我们所说的代数变分，它是代数几何和算术几何研究的中心对象。为了回答这类问题，人们需要检测和计算反映其代数和几何性质的代数变量的各种不变量。本课题研究用于高维变量的不变量，称为零环的Chow群，它可用于代数变量的分类，并涉及多项式方程有理解的存在性问题。根据该奖项的广泛影响，PI将支持一名研究生，并继续她的各种服务和外展活动，包括会议和研讨会组织以及通往弗吉尼亚大学博士课程的桥梁。本项目着重于零环的四个猜想。第一个猜想是关于零环的Chow群的动机过滤的构造。PI先前使用一些k理论技术对阿贝尔变体进行了这一主题的研究，她现在计划将其扩展到K3曲面的某些特殊类别。第二个猜想是由Colliot和Thelene提出的，涉及到p进域上的代数变异，特别是利用各种技术，包括形式群定律和积分p进Hodge理论，建立了关于阿贝尔变异的这个猜想。PI还计划研究两个关于代数数域的猜想，以及它们之间的相容性。其中第一个是Kato和Saito的猜想，可以被认为是零循环的局部到全局原理，她将探索她的工作是否可以构成迈向新型欧拉系统的第一步。第二个猜想是著名的贝林森-布洛赫猜想的一部分，该猜想预测零环的Chow群是有限生成的阿贝尔群。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。

A classical question in number theory is whether a given system of polynomial equations with rational coefficients has a rational solution. Sets of solutions to polynomial equations give rise to what we call algebraic varieties, which are the central object of study in algebraic and arithmetic geometry. In order to answer such types of questions, one needs to detect and compute various invariants of the algebraic variety that reflect its algebraic and geometric properties. This project is concerned with the study of an invariant used for higher dimensional varieties, called the Chow group of zero-cycles, which can be used for classification of algebraic varieties, and relates to the question of existence of rational solutions to polynomial equations. On the broader impact of this award, the PI will support one graduate student and continue her various service and outreach activities including conference and seminar organization and the Bridge to the doctorate program at the University of Virginia. This project focuses on four conjectures for zero-cycles. The first conjecture concerns the construction of motivic filtrations for the Chow group of zero-cycles. The PI has prior work on this subject for abelian varieties using some K-theoretic techniques, which she now plans to extend to some special classes of K3 surfaces. The second conjecture, due to Colliot and Thelene, concerns algebraic varieties over p-adic fields, and in particular establishing this conjecture for abelian varieties using various techniques including formal group laws and integral p-adic Hodge theory. The PI also plans to study two conjectures over algebraic number fields, and the compatibility between them. The first of these is a conjecture of Kato and Saito, which can be thought of as a local-to-global principle for zero-cycles, and she will explore whether her work could constitute the first step towards a new type of Euler system. The second conjecture is part of the famous Beilinson-Bloch conjectures, which predict that the Chow group of zero-cycles is a finitely generated abelian group.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.