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: 2112697
• 负责人: John Doyle
• 金额: $ 11.68万
• 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: 2020
• 资助国家: 美国
• 起止时间: 2020-12-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2112697&HistoricalAwards=false
• 关键词: Moduli; Spaces; Galois; Theory; Arithmetic

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.

算术动力学是一门相对较新的学科，汇集了数学的两个主要领域：数字理论，传统上对多项式方程式的整数研究和整数（或合理的）解决方案以及离散的动态系统，其中一个研究了重复迭代下功能的长期行为。该项目将进一步发展并结合算术动力学的两个方面：一种本质上是几何的，使用几何对象（模量空间）将动态系统分类为具有指定的动力学行为，而另一个是代数，理解代数对称性（GALOIS理论）与动态系统有关。数量理论的领域以及在某种程度上发现了算术动力学，在密码学和相关领域中发现了用途。 PI将继续进行外展活动，目的是使用密码学作为向更普遍的受众介绍有趣数学的一种手段。该项目由代数和数理论计划以及刺激竞争研究的既定计划共同资助。一种重要的联系是，用于投射空间内态的前观点起着与椭圆曲线上的扭转点相似的作用（或更一般而言，是Abelian品种）。该项目的重点是从模量理论和galois理论的角度进一步研究这种类比。 PI参与了模量空间理论的发展，该理论具有标记前观点的内态性，类似于经典模块化曲线，该曲线类似于经典的模块化曲线，该曲线具有带有明显的扭转点的椭圆形曲线。这样的模量空间已经在莫顿和西尔弗曼的动态统一界面构想中发挥了基本作用，莫顿和西尔弗曼是Mazur-Merel强统一界定理的动力学类似物，用于旋转点的椭圆形曲线。为了在这个困难的统一界面问题上取得进一步的进展，PI提议研究附着在某些动态有趣的功能家族的动力模量空间的几何形状（例如，具有给定时期关键点的二次理性图）。前观点点和扭转点之间的类比也使自己成为Serre开放图像定理的动态类似物，这是与椭圆曲线上与扭转点相关的Adelic Galois表示的有限索引结果。 PI提议研究（PRE）定期为理性地图的定期点附加的适当的GALOIS表示形式 - 尤其是在功能领域设置中，结果将为动态模量空间的几何形状提供新的见解。该奖项反映了NSF的法定任务，并通过使用基金会的知识优点和广泛的criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia rection the Apportiation。

项目成果

New families satisfying the dynamical uniform boundedness principle over function fields

满足函数域上动态一致有界原理的新族

• DOI: 10.1007/s00208-022-02536-z
• 发表时间: 2022
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Doyle, John R.;Faber, Xander
• 通讯作者: Faber, Xander

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

动态多项式、项链算子和动力单位的通用关系

• 发表时间: 2022
• 期刊: New York journal of mathematics
• 影响因子: 0.6
• 作者: Doyle, John R.;Fili, Paul;Hyde, Trevor
• 通讯作者: Hyde, Trevor

Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations

动态模空间和构型的多项式自同态

• DOI: 10.1007/s40598-022-00197-z
• 发表时间: 2022
• 期刊: Arnold Mathematical Journal
• 作者: Blum, Talia;Doyle, John R.;Hyde, Trevor;Kelln, Colby;Talbott, Henry;Weinreich, Max
• 通讯作者: Weinreich, Max

Multivariate polynomial values in difference sets

差异集中的多元多项式值

• 发表时间: 2021
• 期刊: Discrete analysis
• 影响因子: 1.1
• 作者: Doyle, John R.;Rice, Alex
• 通讯作者: Rice, Alex