CAREER: Arithmetic Dynamical Systems on Projective Varieties

这个项目的中心是最近一个新的数学领域的问题，称为算术动力学。这门学科综合了以前完全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方面的力量，以及动力学作为引人注目的算术问题的来源的丰富性。这个项目的资金将通过课程和研讨会等活动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，以及PI和相邻领域的研究人员之间的合作。该项目的第一个重点领域是阿贝尔变种的设置，PI计划解决围绕定义和S领域的各种猜想-小规范高度点的完整性。这项研究的一个重要组成部分是得到了广义Arakelov-Green函数平均值的定量下界，推广了已有的一维情形下的结果。PI旨在为任意极化动力系统发展这样的结果，为广泛的算术应用开辟一条道路。第二个焦点领域涉及数域和一维函数域上的曲线上的Arakelov不变量与它们的雅可比变体上的算术之间的关系。本文的目的是将张氏可容许相对对偶化层的自交与雅可比上的小点运算以及其他显著的Arakelov不变量，如Delta不变量联系起来。第三个目标是研究亚种的典范高度，特别是在有因子的情况下。这里主要关注的是动力系统复杂性的各种衡量标准与某些亚类的高度之间的关系。该项目的最后部分旨在将上述广义阿拉克洛夫-格林函数与多势理论联系起来，既有复杂的，也有非阿基米德的。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.