Moduli Spaces and Galois Theory in Arithmetic Dynamics

算术动力学中的模空间和伽罗瓦理论

• 批准号: 2302394
• 负责人: John Doyle
• 金额: $ 13.65万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-15 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302394&HistoricalAwards=false
• 关键词: Moduli; Spaces; Galois; Theory; Arithmetic

Arithmetic dynamics is a quickly growing area of mathematics that combines ideas from several disciplines: number theory, which is typically concerned with properties of the integers, including prime numbers; arithmetic geometry, whose primary goal is to understand integer and rational solutions to systems of Diophantine equations; and dynamics, which is the study of how systems change over time. This project will approach problems in arithmetic dynamics from two different directions: One is a geometric approach, studying dynamical moduli spaces -- geometric objects that classify dynamical systems that have various interesting dynamical behaviors. The other is an algebraic approach, understanding various algebraic symmetries exhibited by dynamical systems and the way that these symmetries interact with dynamical moduli spaces. In addition to working with graduate students on problems in arithmetic dynamics, this project will involve outreach in the community, to middle and high school students as well as adults working to further their education.Arithmetic dynamics is largely motivated by analogies between objects in arithmetic geometry and the dynamics of rational maps. One explicit connection is that preperiodic points for rational functions form a natural dynamical analogue of torsion points on elliptic curves. To better understand the torsion points on elliptic curves, one is led to consider modular curves, which parametrize (isomorphism classes of) elliptic curves together with level structure, a key example of which would be marking a torsion point of order n. In a similar fashion, one approach to studying algebraic dynamics is to consider moduli spaces for dynamical systems with a dynamical notion of level structure: for example, one might study the (equivalence classes) of rational functions of a given degree d together with a marked periodic point of period n. The PI will continue his work developing these dynamical moduli spaces and better understanding geometric, arithmetic, and Galois-theoretic properties. Work on this project will lead to insights into two directions in arithmetic dynamics: The Morton-Silverman dynamical uniform boundedness conjecture, which is a strengthening of Merel's theorem for torsion points on elliptic curves, and dynamical analogues of Serre’s open image theorem.This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

算术动力学是一个快速发展的数学领域，它结合了多个学科的思想：数论，通常关注整数的属性，包括素数；算术几何，其主要目标是理解系统的整数和有理解；丢番图方程；以及动力学，研究系统如何随时间变化。该项目将从两个不同的方向处理算术动力学问题：一个是几何方法，研究动态模空间——对动力系统进行分类的几何对象。另一个是代数方法，了解动力系统表现出的各种代数对称性以及这些对称性与动力模空间相互作用的方式，除了与研究生一起研究算术动力学问题之外。涉及到社区、中学生以及从事教育工作的成年人。算术动力学很大程度上是由算术几何中的对象与进一步的有理图的动力学之间的类比所激发的。有理函数的前周期点形成了椭圆曲线上扭转点的自然动力学模拟。为了更好地理解椭圆曲线上的扭转点，我们考虑模曲线，它将椭圆曲线（的同构类）与水平结构一起参数化。 ，其中一个关键的例子是标记 n 阶扭转点。以类似的方式，研究代数动力学的一种方法是考虑动力学的模空间。具有能级结构动态概念的系统：例如，可以研究给定次数 d 的有理函数（等价类）以及周期 n 的标记周期点。 PI 将继续开发这些动态模空间和更好地理解几何、算术和伽罗瓦理论性质，该项目的工作将深入了解算术动力学的两个方向：莫顿-西尔弗曼动力学一致有界猜想，它是对椭圆曲线上的扭点梅雷尔定理，以及塞尔开图像定理的动力学类似物。该项目由代数和数论计划以及刺激竞争性研究既定计划 (EPSCoR) 联合资助。该奖项反映了 NSF 的法定使命，并具有通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。