Arakelov Geometry and Algebraic Dynamics

这个奖项的项目在于代数动力学和算术几何的接口，其动机是投影簇和动力系统轨道上的合理点之间的类比。理解多项式方程的有理解是一个经典和基本的问题，费马大定理和莫德尔猜想的证明是现代数学进步中最著名的例子之一。这些问题在动力系统领域找到了类似物以及更广泛的框架。通过该提案资助的项目侧重于在数域和函数域上定义的投影变种上的动力系统。一个关键的目的是扩大知识的算术点的小正则高度相对于合理的功能，更一般的极化动力系统。该项目的资金将支持UIC一个不断增长的小组的基础设施，该小组专门研究数论，动力学和逻辑的交叉问题。它还将支持PI与其他研究人员之间的合作，这些研究人员的工作将非阿基米德分析和阿拉克洛夫几何应用于数论问题。PI计划在BIRS组织一次研讨会，并成为未来妇女参与数字研究小组的项目领导人。第一部分研究了维数大于1的极化动力系统小正则高度点的计算。具体的方向包括扭转猜想的阿贝尔品种，沿着与其功能领域的模拟，以及稀疏性猜想扭转点的阿贝尔品种是S-积分相对于非扭转充分因子。在这个标题下也福尔斯一个项目研究某些杰出的地方规范高度的阿贝尔品种和他们的关系，全球Neron-Tate高度。第二个线程连接Arakelov不变量在更高的亏格曲线的关键丢番图关于他们的合理的点，利用最近的工作，这些不变量的动力学和雅可比品种的分析。这条线索的一个中心部分集中在张所介绍的容许相对对偶层的自相交上，并将这个量与底层曲线的度量不变量联系起来。第三个项目是关于与投射簇相关的Berkovich解析空间的多能理论。在这里，一个特定的最终目标是为任意维度的小点开发合适的定量等分布声明，从而自然地应用于上述第一个项目。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智力价值和更广泛的影响审查标准进行评估来支持。

The project in this award lies at the interface of algebraic dynamics and arithmetic geometry, motivated by an analogy between rational points on projective varieties and orbits of dynamical systems. Understanding the rational solutions to polynomial equations is a classical and fundamental problem, with advances such as Fermat's Last Theorem and the proof of the Mordell Conjecture being among the most celebrated examples of progress in modern mathematics. These questions find analogues as well as broader frameworks in the area of dynamical systems. The project funded through this proposal focuses on dynamical systems on projective varieties defined over number fields and function fields. A key aim is to extend knowledge on the arithmetic of points of small canonical height with respect to rational functions to more general polarized dynamical systems. The funding for this project will support the infrastructure of a growing group at UIC specializing in questions at the intersection of number theory, dynamics, and logic. It will also support collaborations between the PI and other researchers whose work applies non-archimedean analysis and Arakelov geometry to number-theoretic problems. The PI plans to organize a workshop at BIRS and to be a project leader for a future Women in Numbers research team.Three types of problems will be investigated. The first centers on the arithmetic of points of small canonical height for polarized dynamical systems in dimension larger than 1. Specific directions include the Torsion Conjecture for abelian varieties, along with its function field analogue, as well as a sparseness conjecture for torsion points on abelian varieties that are S-integral relative to a non-torsion ample divisor. Under this heading also falls a project studying certain distinguished local canonical heights on abelian varieties and their relation to the global Neron-Tate height. The second thread connects Arakelov invariants on higher genus curves to key Diophantine conjectures about their rational points, capitalizing on recent work linking these invariants to dynamics and analysis on Jacobian varieties. A central component of this thread focuses on the self-intersection of the admissible relative dualizing sheaf introduced by Zhang, and links this quantity to metric invariants of the underlying curve. The third project concerns pluripotential theory on Berkovich analytic spaces associated to projective varieties. Here a particular ultimate goal is the development of suitable quantitative equidistribution statements for small points in arbitrary dimension, yielding natural applications to the first aforementioned project.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.