Moduli Spaces and Galois Theory in Arithmetic Dynamics

算术动力学是一门相对较新的学科，它汇集了数学的两个主要领域：数论，传统上研究多项式方程的整数和整数（或有理数）解；离散动力系统，研究函数在重复迭代下的长期行为。该项目将进一步发展和结合算术动力学的两个方面：一个是几何性质的，使用几何对象（模空间）对具有特定动力行为的动力系统进行分类，另一个是代数的，理解与动力系统相关的代数对称性（伽罗瓦理论）。数论领域，在某种程度上，算术动力学在密码学和相关领域都有应用。PI将继续开展外展活动，目的是使用密码学作为向更广泛的受众介绍有趣的数学的手段。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。算术动力学领域在很大程度上是由算术几何和动力系统之间的类比所推动的。其中一个重要的联系是，射影空间自同态的周期前点的作用类似于椭圆曲线上的扭转点（或者更一般地说，是阿贝尔变体）。本课题的重点是从模理论和伽罗瓦理论的角度进一步研究这一类比。PI参与了模数空间理论的发展，该理论将带有标记的周期前点的自同态参数化——类似于经典模曲线，它将带有标记的扭转点的椭圆曲线参数化。这样的模空间已经在Morton和Silverman的动力学一致有界性猜想的进展中发挥了基本作用，这是椭圆曲线上扭转点的Mazur-Merel强一致有界性定理的动力学模拟。为了在这一困难的一致有界性问题上取得进一步的进展，PI建议研究附属于某些动态有趣的函数族（例如，具有给定周期临界点的二次有理映射）的动态模空间的几何。前周期点和扭转点之间的类比也适用于Serre开像定理的动态模拟，Serre开像定理是椭圆曲线上与扭转点相关的阿得利克格罗瓦表示的有限指数结果。PI建议研究合理映射的（预）周期点的伽罗瓦表示法——特别是在函数域设置中，其结果将为动态模空间的几何提供新的见解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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 relatively new discipline that brings together two major areas of mathematics: number theory, traditionally the study of the integers and integer (or rational) solutions to polynomial equations, and discrete dynamical systems, where one studies the long term behavior of functions under repeated iteration. This project will further develop and combine two facets of arithmetic dynamics: one is geometric in nature, using geometric objects (moduli spaces) to classify dynamical systems with specified dynamical behaviors, and the other is algebraic, understanding algebraic symmetries (Galois theory) associated to dynamical systems. The fields of number theory and, to some extent, arithmetic dynamics have found uses in cryptography and related areas. The PI will continue outreach activities with the aim of using cryptography as a means of introducing a more general audience to interesting mathematics. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).The field of arithmetic dynamics is heavily motivated by analogies between arithmetic geometry and dynamical systems. One important such connection is that preperiodic points for endomorphisms of projective space play a role similar to torsion points on elliptic curves (or, more generally, abelian varieties). The focus of this project is to further investigate this analogy from the moduli-theoretic and Galois-theoretic perspectives. The PI has been involved with the development of the theory of moduli spaces that parametrize endomorphisms with marked preperiodic points -- analogous to classical modular curves, which parametrize elliptic curves with marked torsion points. Such moduli spaces have already played a fundamental role in progress toward the dynamical uniform boundedness conjecture of Morton and Silverman, a dynamical analogue of the Mazur-Merel strong uniform boundedness theorem for torsion points on elliptic curves. In order to make further progress on this difficult uniform boundedness problem, the PI proposes to study the geometry of dynamical moduli spaces attached to certain dynamically interesting families of functions (e.g., quadratic rational maps with a critical point of a given period). The analogy between preperiodic points and torsion points also lends itself to a dynamical analogue of Serre's open image theorem, a finite-index result for the adelic Galois representation associated to torsion points on elliptic curves. The PI proposes studying the appropriate Galois representation attached to (pre)periodic points for rational maps -- especially in the function field setting, where results will provide new insights into the geometry of dynamical moduli spaces.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.