CAREER: Arithmetic Dynamical Systems on Projective Varieties

职业：射影簇的算术动力系统

• 批准号: 2337942
• 负责人: Nicole Looper
• 金额: $ 40万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2029-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2337942&HistoricalAwards=false
• 关键词: CAREER; Arithmetic; Dynamical; Systems; Projective

This project centers on problems in a recent new area of mathematics called arithmetic dynamics. This subject synthesizes problems and techniques from the previously disparate areas of number theory and dynamical systems. Motivations for further study of this subject include the power of dynamical techniques in approaching problems in arithmetic geometry and the richness of dynamics as a source of compelling problems in arithmetic. The funding for this project will support the training of graduate students and early career researchers in arithmetic dynamics through activities such as courses and workshops, as well as collaboration between the PI and researchers in adjacent fields.The project’s first area of focus is the setting of abelian varieties, where the PI plans to tackle various conjectures surrounding the fields of definition and S-integrality of points of small canonical height. One important component of this study is the development of quantitative lower bounds on average values of generalized Arakelov-Green’s functions, which extend prior results in the dimension one case. The PI intends to develop such results for arbitrary polarized dynamical systems, opening an avenue for a wide variety of arithmetic applications. A second area of focus concerns the relationship between Arakelov invariants on curves over number fields and one-dimensional function fields, and arithmetic on their Jacobian varieties. Here the project aims to relate the self-intersection of Zhang’s admissible relative dualizing sheaf to the arithmetic of small points on Jacobians, as well as to other salient Arakelov invariants such as the delta invariant. The third goal is to study canonical heights of subvarieties, especially in the case of divisors. A main focus here is the relationship between various measurements of the complexity of the dynamical system and the heights of certain subvarieties. The final component of the project aims to relate the aforementioned generalized Arakelov-Green’s functions topluripotential theory, both complex and non-archimedean.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.

该项目的重点是最近一个新的数学领域（称为算术动力学）中的问题。这个主题综合了以前数论和动力系统不同领域的问题和技术。进一步研究这一主题的动机包括动力技术在算术几何中接近问题的力量，以及动力学作为算术中引人注目的问题的来源的丰富性。该项目的资金将通过课程和研讨会等活动，以及PI与相邻领域研究人员之间的合作，支持研究生和早期职业研究人员在算术动力学方面的培训。该项目的第一个重点领域是阿贝尔品种的设置，其中PI计划解决围绕小规范高度点的定义和S-积分领域的各种问题。本研究的一个重要组成部分是广义Arakelov-Green函数的平均值的定量下限的发展，扩展了以前的结果在一维的情况下。PI打算为任意极化动力系统开发这样的结果，为各种各样的算术应用开辟了一条道路。第二个领域的重点关注的关系Arakelov不变量的曲线在数域和一维函数领域，算术的雅可比品种。在这里，该项目的目的是将张的容许相对对偶层的自相交与雅可比行列式上的小点的算术以及其他突出的阿拉克洛夫不变量（如δ不变量）联系起来。第三个目标是研究子簇的典范高度，特别是在因子的情况下。这里的一个主要焦点是各种测量的复杂性的动力系统和高度的某些子品种之间的关系。该项目的最后一部分旨在将上述广义的阿拉克洛夫-格林函数与复杂的非阿基米德多能理论联系起来。该奖项反映了NSF的法定使命，并被认为值得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来支持。