Investigations in Arithmetic Geometry and Arithmetic Dynamics

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，该猜想描述了某些放大变种上有理点的大小。算术几何中的第二个项目与Heegner在虚二次场相关的模椭圆曲线上构造特殊点以及Deuring的一个经典结果有关，该结果解释了如何将“mod p”点提升到Heegner点。最近，Darmon展示了如何构造heegner型点，而不是与实二次域相关联。PI将研究这些damon - heegner点的deuring型提升结果的可能性。PI的其他项目在算术动力学领域，这是一个新的领域，研究与多项式或有理函数迭代相关的离散动力系统的代数、数论和p进性质。PI计划在这一领域研究四个问题：(1)正则仿射自同构下高度函数的变换性质。(2) Mandelbot集中Misiurewicz点的算术性质。(3)射影空间和其他射影变体上的p进动力学和非阿基米德格林函数。(4)有限特征的栅格图分类。用整数或有理数解多项式方程，自古以来就有人研究过。保罗·沃伊塔（Paul Vojta）在20世纪80年代提出了一个基本猜想，从几何角度描述了这些解的大小。PI计划用新的方程类来证明Vojta的猜想。椭圆曲线是由一类特殊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“模p”解提升到实际解。这些提升的解被称为Heegner点。最近，Darmon构造了一种新的Heegner点。PI将研究将“mod p”解提升到这些新的damon - heegner点的方法。PI的另一个研究领域是算术动力学。动力系统研究的是当一个函数迭代时不同的起始点会发生什么，也就是说，取一个函数f(x)和一个起始点b，观察序列f(b), f(f(b)), f(f(b))),... .算术动力学的新领域要求这些迭代值具有数论性质。PI将研究算术动力学中的问题，包括研究迭代值的复杂性，著名的Mandelbrot集合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迭代的问题。

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.