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）解，以及离散动力系统，其中研究函数在重复迭代下的长期行为。本项目将进一步发展和联合收割机的两个方面的算术动力学：一个是几何的性质，使用几何对象（模空间）来分类具有指定的动力学行为的动力系统，另一个是代数，理解代数对称（伽罗瓦理论）与动力系统。数论领域，在某种程度上，算术动力学已经在密码学和相关领域中得到了应用。PI将继续开展外联活动，目的是利用密码学作为向更广泛的受众介绍有趣的数学的一种手段。该项目由代数和数论计划和刺激竞争研究的既定计划（EPSCoR）共同资助。算术动力学领域受到算术几何和动力系统之间类比的极大推动。一个重要的联系是，射影空间自同态的预周期点扮演着类似于椭圆曲线（或者更一般地说，阿贝尔簇）上扭点的角色。本计画的重点是从模论与伽罗瓦理论的观点进一步探讨此类比。PI参与了模空间理论的发展，该理论用标记的预周期点参数化自同态-类似于经典的模曲线，用标记的扭点参数化椭圆曲线。这样的模空间已经在莫顿和西尔弗曼的动力学一致有界性猜想的进展中发挥了重要作用，该猜想是椭圆曲线上扭点的Mazur-Merel强一致有界性定理的动力学模拟。为了在这个困难的一致有界性问题上取得进一步的进展，PI建议研究与某些动态有趣的函数族（例如，具有给定周期的临界点的二次有理映射）。前周期点和扭点之间的类比也适用于塞尔开像定理的动力学模拟，这是椭圆曲线上与扭点相关的椭圆伽罗瓦表示的有限指数结果。PI建议研究适当的伽罗瓦表示附加到（前）周期点的合理地图-特别是在功能字段设置，其中的结果将提供新的见解动态moduli spaces.This奖项的几何形状反映了NSF的法定使命，并已被认为是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的支持。

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.