CAREER: Arithmetic Dynamical Systems on Projective Varieties

这个项目的中心问题在一个新的数学领域称为算术动力学。这门学科综合了以前不同的数论和动力系统领域的问题和技术。进一步研究这一主题的动机包括动力学技术在处理算术几何问题中的力量，以及动力学作为算术中引人注目的问题的丰富来源。该项目的资金将通过课程和研讨会等活动，以及PI与邻近领域的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变量的设置，PI计划解决围绕小规范高度点的定义和s -完整性领域的各种猜想。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界的发展，它推广了先前在一维情况下的结果。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一条道路。第二个关注的领域是数域和一维函数域上曲线上的Arakelov不变量之间的关系，以及它们的雅可比变量的算术。在这里，该项目旨在将张的可容许相对对偶束的自交与雅可比矩阵上小点的算法以及其他显著的Arakelov不变量（如delta不变量）联系起来。第三个目标是研究子变体的正则高度，特别是在除数的情况下。这里的主要焦点是动力系统复杂性的各种测量与某些子变量的高度之间的关系。该项目的最后一个组成部分旨在将上述广义Arakelov-Green函数与复数势理论联系起来，包括复杂的和非阿基米德的。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。

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.