Arakelov Geometry and Algebraic Dynamics

这个奖项的项目位于代数动力学和算术几何的交界处，其动机是射影簇上的有理点和动力系统的轨道之间的类比。理解多项式方程的有理数解是一个经典和基本的问题，费马最后定理和莫德尔猜想的证明等进展是现代数学进步的最著名的例子之一。这些问题在动力系统领域找到了相似之处和更广泛的框架。通过这项提议资助的项目侧重于在数域和函数域上定义的射影变种的动力系统。一个关键的目的是将关于关于有理函数的小正则高度点的算术的知识推广到更一般的极化动力系统。这个项目的资金将支持UIC一个不断壮大的小组的基础设施，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和其他研究人员之间的合作，这些研究人员的工作是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组织一个研讨会，并成为未来女性数字研究小组的项目负责人。将调查三种类型的问题。第一个中心是关于维大于1的极化动力系统的小正则高度点的计算。具体方向包括关于阿贝尔变种的扭转猜想及其函数场模拟，以及关于阿贝尔变种的扭点的一个稀疏猜想，该猜想相对于一个非挠足因子是S积分的。在这个标题下还有一个项目，研究阿贝尔品种的某些独特的地方典范高度及其与全球农伦塔高度的关系。第二条线索将高亏格曲线上的Arakelov不变量与关于其有理点的关键丢番图猜想联系起来，利用最近将这些不变量与Jacobian簇的动力学和分析联系起来的工作。这条线索的一个中心成分集中在张引入的容许相对对偶化层的自交上，并将这个量与基础曲线的度量不变量联系起来。第三个项目涉及与射影簇相关的Berkovich分析空间的多势理论。在这里，一个特别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，从而自然地应用于第一个上述项目。该奖项反映了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。

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.