Algebraic Dynamics over Global Fields: Geometric and Analytic Methods

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”本研究项目的目的是研究全局和局部领域的代数动力学。本建议中详细介绍的一些具体目标与该学科中众所周知的开放问题有关，包括Ih关于积分周期前点的猜想，Morton-Silverman一致有界猜想的函数场模拟，以及椭圆曲线上Szpiro猜想的动态模拟的公式。所提出的技术包括使用Berkovich解析空间、规范高度函数、等分布、经典和非阿基米德法图-茱莉亚理论、自同态的模空间、阿德尔容量理论和丢芬图不等式。从广义上讲，这个提议中的大多数思想可以看作是两个一般主题的反映：第一是曲线的数场和函数场之间的强平行，第二是通过使用Berkovich解析空间统一处理阿基米德和非阿基米德代数动力学。代数动力学是一门将经典动力系统的要素与代数几何相结合的学科。它的基本研究对象是一个表示某些代数方程解的集合X，以及一个从X到它自身的函数f。说这些对象是在全局和局部域上定义的，本质上意味着它们对数论学家和算术几何学家特别感兴趣。当一个人反复迭代函数f时，这个主题的动态方面就开始发挥作用了。例如，如果x是集合x的一个点，对于这个点，无限序列f(x), f(f(x)), f(f(f(x))), f(f(f(x)))))，…在代数动力学中，许多最基本的问题都涉及到预周期点的性质，但它们仍然没有被很好地理解。本研究计划旨在解决的自然开放问题的一些例子包括：1。有多少个周期前点是代数整数？2. 给定两个不同的函数f和g从X到自身，关于这两个函数有多少点是周期前的？3. 一个周期前点序列接近一个给定的非周期前点的速度有多快？

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The aim of this research project is to study algebraic dynamics over global and local fields. Some of the specific goals detailed in this proposal are related to well-known open problems in the subject, including Ih's conjecture on integral preperiodic points, a function field analogue of Morton-Silverman's uniform boundedness conjecture, and the formulation of a dynamical analogue of Szpiro's conjecture for elliptic curves. The proposed techniques include the use of Berkovich analytic spaces, canonical height functions, equidistribution, classical and non-archimedean Fatou-Julia theory, moduli spaces of endomorphisms, adelic capacity theory, and Diophantine inequalities. Broadly, the majority of the ideas in this proposal can be viewed as reflections of two general themes: the first is the strong parallel between number fields and function fields of curves, and the second is the unified treatment of archimedean and non-archimedean algebraic dynamics, through the use of Berkovich analytic spaces. Algebraic dynamics is a subject which combines elements of classical dynamical systems with algebraic geometry. Its basic object of study is a set X representing the solutions to certain algebraic equations, together with a function f from X into itself. To say that these objects are defined over global and local fields means essentially that they are of special interest to number theorists and arithmetic geometers. The dynamical aspect of the subject comes into play when one repeatedly iterates the function f. For example, if x is a point of the set X for which the infinite sequence f(x), f(f(x)), f(f(f(x))), f(f(f(f(x)))), ..., eventually falls into a finite loop, then x is called a preperiodic point with respect to f. Many of the most fundamental questions in algebraic dynamics involve the nature of preperiodic points, and yet they are still not very well understood. Some examples of natural open questions which this research proposal aims to address include the following: 1. How many preperiodic points are algebraic integers? 2. Given two different functions f and g from X into itself, how many points are preperiodic with respect to both functions? 3. How rapidly can a sequence of preperiodic points approach a given non-preperiodic point?