Investigations in Arithmetic Geometry and Arithmetic Dynamics

算术几何和算术动力学研究

• 批准号: 0650017
• 负责人: Joseph Silverman
• 金额: $ 19.48万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0650017&HistoricalAwards=false
• 关键词: Investigations; Arithmetic; Geometry; Dynamics

The PI will investigate two problems in arithmetic geometry and several problems in arithmetic dynamics. In the first arithmetic geometry project, the PI will prove new cases of a fundamental conjecture of Paul Vojta describing the size of rational points on certain blowup varieties. The second project in arithmetic geometry is related to Heegner's construction of special points on modular elliptic curves associated to imaginary quadratic fields and to a classical result of Deuring that explains how to lift "mod p" points to Heegner points. Recently Darmon showed how one might construct Heegner-type points associated instead to real quadratic fields. The PI will investigate the possibility of a Deuring-type lifting result for these Darmon-Heegner points. The PI's other projects are in the area of arithmetic dynamics, which is a new field in which one studies algebraic, number theoretic, and p-adic properties of discrete dynamical systems associated to iteration of polynomial or rational functions. The PI plans to investigate four problems in this area: (1) Transformation properties of height functions under regular affine automorphisms. (2) Arithmetic properties of Misiurewicz points in the Mandelbot set. (3) p-adic dynamics and nonarchimedean Green functions on projective space and other projective varieties. (4) Classification of Latths maps in finite characteristic.The solution of polynomial equations using integers or rational numbers has been studied since antiquity. A fundamental conjecture of Paul Vojta from the 1980's describes the size of such solutions in terms of geometry. The PI plans to prove Vojta's conjecture for new classes of equations. Elliptic curves, which are defined by a particular type of polynomial equation, have been extensively studied during the past 80 years. In the 1930's, Deuring described how to lift certain "mod p" solutions to actual solutions. These lifted solutions are called Heegner points. Recently Darmon constructed a new type of Heegner point. The PI will study ways to lift "mod p"solutions to these new Darmon-Heegner points. The PI's other area of research is in the field of arithmetic dynamics. Dynamical systems is the study of what happens to different starting points when a function is iterated, i.e., take a function f(x) and a starting point b and look at the sequence f(b), f(f(b)), f(f(f(b))),... . The new field of arithmetic dynamics asks for number theoretic properties of these iterated values. The PI will investigate problems in arithmetic dynamics, including studying the complexity of the iterated values, number theoretic properties of certain special points in the famous Mandelbrot set, and problems involving iteration of certain functions defined using elliptic curves.

PI将研究算术几何中的两个问题和算术动力学中的几个问题。 在第一个算术几何项目中，PI将证明Paul Vojta的一个基本猜想的新情况，该猜想描述了某些爆破品种上的有理点的大小。第二个项目在算术几何是有关Heegner的建设特殊点的模块化椭圆曲线相关联的虚二次领域和一个经典的结果杜林解释如何解除“模p”点Heegner点。最近达蒙表明，人们可能如何建设希格纳型点，而不是相关的真实的二次领域。PI将调查这些Darmon-Heegner点出现Deuring类型提升结果的可能性。PI的其他项目是在算术动力学领域，这是一个新的领域，其中一个研究代数，数论和p-进性质的离散动力系统相关的多项式或有理函数的迭代。PI计划研究这一领域的四个问题：（1）高度函数在正则仿射自同构下的变换性质。(2)Mandelbot集合中Misiurewicz点的算术性质。(3)p-adic动力学和非阿基米德绿色函数在射影空间和其他射影簇。(4)有限特征格映射的分类自古以来，人们就研究用整数或有理数解多项式方程。Paul Vojta在20世纪80年代的一个基本猜想描述了这种解的几何尺寸。PI计划证明Vojta猜想的新类别的方程。 椭圆曲线是由一种特殊类型的多项式方程定义的，在过去的80年里得到了广泛的研究。 在20世纪30年代，杜林描述了如何将某些“mod p”解提升为实际解。这些提升的解称为Heegner点。 最近Darmon构造了一种新的Heegner点。PI将研究如何将“mod p”解决方案提升到这些新的Darmon-Heegner点。 PI的其他研究领域是算术动力学领域。 动力系统是研究当一个函数迭代时，不同的起始点会发生什么，即，取一个函数f（x）和一个起点B，看看序列f（B），f（f（B）），f（f（f（B），. .算术动力学的新领域要求这些迭代值的数论性质。 PI将研究算术动力学中的问题，包括研究迭代值的复杂性，着名的Mandelbrot集中某些特殊点的数论性质，以及涉及使用椭圆曲线定义的某些函数的迭代的问题。