Zero-cycles over local and global fields

局部和全局领域的零循环

• 批准号: 2302196
• 负责人: Evangelia Gazaki
• 金额: $ 18.07万
• 依托单位: University of Virginia Main Campus
• 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: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302196&HistoricalAwards=false
• 关键词: Zero; cycles; over; local; global

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.

数论中的一个经典问题是给定的有理系数多项式方程组是否有有理解。多项式方程的解集产生了我们所说的代数簇，它是代数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代数簇的各种不变量，这些不变量反映了代数簇的代数和几何性质。该项目关注的是用于高维簇的不变量的研究，称为零圈的Chow群，可用于代数簇的分类，并涉及多项式方程有理解的存在性问题。在这个奖项的更广泛的影响，PI将支持一名研究生，并继续她的各种服务和推广活动，包括会议和研讨会组织和桥梁博士学位课程在弗吉尼亚大学。本项目重点研究了零周期的四种实现方法。第一个猜想是关于零圈Chow群的motivic滤子的构造。PI已经在这个问题上使用一些K-理论技术的阿贝尔品种，她现在计划扩展到一些特殊类的K3表面的先前工作。第二个猜想，由于Colliot和Thelene，涉及代数品种的p-adic领域，特别是建立这一猜想的阿贝尔品种使用各种技术，包括正式的群体法律和积分p-adic霍奇理论。PI还计划研究代数数域上的两个代数，以及它们之间的兼容性。其中第一个是加藤和齐藤的猜想，这可以被认为是零圈的局部到全局原理，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Beilinson-Bloch猜想的一部分，该猜想预测了Chow群的零圈是一个由零圈生成的阿贝尔群。该奖项反映了NSF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。