Moduli Spaces and Galois Theory in Arithmetic Dynamics

算术动力学是一个迅速发展的数学领域，它融合了几个学科的思想：数论，它通常涉及整数的性质，包括质数；算术几何，它的主要目标是了解丢番图方程组的整数和有理解；以及动力学，它研究系统如何随着时间的变化。这个项目将从两个不同的方向来研究算术动力学中的问题：一个是几何方法，研究动力模空间--对具有各种有趣的动力学行为的动力系统进行分类的几何对象。另一种是代数方法，了解动力系统所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的问题外，这个项目还将在社区、初中生和成人中展开工作，以进一步提高他们的教育水平。算术动力学在很大程度上是由算术几何中的对象和有理映射的动力学之间的类比所推动的。一个明显的联系是，有理函数的准周期点形成了椭圆曲线上扭点的自然动力学模拟。为了更好地理解椭圆曲线上的扭点，人们考虑了模曲线，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阶的扭点。以类似的方式，研究代数动力学的一种方法是考虑具有水平结构的动力学概念的动力系统的模空间：例如，人们可以研究给定次数d的有理函数的(等价类)以及周期为n的显著周期点。PI将继续他的工作，开发这些动态模空间，并更好地理解几何、算术和伽罗瓦理论的性质。这个项目的工作将导致对算术动力学两个方向的深入研究：Morton-Silverman动态一致有界性猜想，它是对椭圆曲线上扭点的Merel定理的加强，以及Serre开放映象定理的动态类似。该项目由代数和数论计划和既定的刺激竞争研究计划(EPSCoR)联合资助。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。

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.