Moduli Spaces and Galois Theory in Arithmetic Dynamics

算术动力学是一个相对较新的学科，它汇集了两个主要的数学领域：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，研究函数在重复迭代下的长期行为。这个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)来分类具有特定动力学行为的动力系统；另一个是代数，了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，算术动力学已经在密码学和相关领域找到了用处。国际数学联合会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。这个项目是由代数和数论计划和建立的激励竞争研究计划(EPSCoR)共同资助的。算术动力学领域主要是由算术几何和动力系统之间的类比推动的。其中一个重要的联系是，射影空间的自同态的准周期点起着类似于椭圆曲线上的扭点的作用(或者，更一般地，阿贝尔簇)。本项目的重点是从模理论和伽罗瓦理论的角度进一步研究这种类比。PI参与了模空间理论的发展，它用标记的预周期点来参数化自同态--类似于经典的模曲线，它用标记的扭点来参数化椭圆曲线。这种模空间已经在Morton和Silverman的动力一致有界性猜想的发展过程中发挥了基本作用，该猜想是对椭圆曲线上扭点的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)上的动力模空间的几何。准周期点和扭点之间的类比也适用于Serre开象定理的动力学类比，该定理是与椭圆曲线上的扭点相关的有限指数伽罗瓦表示的结果。PI建议研究有理映射的(准)周期点上的适当伽罗瓦表示--特别是在函数域环境中，其中的结果将为动态模空间的几何提供新的见解。这一奖项反映了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 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.