Applications of arithmetic dynamics to potential density

这个项目处理的问题是确定在数字场上定义的射影变量X何时具有所谓的势密度性质。也就是说，确定是否存在我们的数字域的有限扩展，使得X在这个更大的数字域上有许多点（或者，用更专业的语言来说，一个Zariski密集的）集合。虽然这个问题是用几何语言表述的，但它的历史悠久，比现代代数几何早了数百年。在最早的形式中，我们可以把诸如求出所有毕达哥拉斯三元组或求出佩尔方程的解的问题看作是这个问题的变体，在这些情况下，我们研究的变量是在有理数上定义的曲线，我们试图找到许多不同的整数解。**第一个有趣但尚未完全理解的情况是理解表面和三倍的潜在密度。这里已经有了很多已知的，并且由于Castelnuovo和Enriques，有一个粗略的表面分类，这是很大的帮助。这种分类表明，在一些等价的概念下，曲面可以根据某些不变量进行分类。根据这种分类，许多类别的表面的势密度特性是完全理解的。在某些情况下，势密度只能通过附加信息来理解。这方面的一个例子是Bogomolov和Tschinkel最近的工作，他们表明，对于一类称为K3曲面，人们可以证明，如果自同构群是无限的，则势密度性质成立。变种的自同构群在某种意义上描述了它的对称性所以说变种有一个大的自同构群就是在说它不是那么刚性。****我的目标是以Bogomolov和Tschinkel的工作为起点，研究具有无限自同构群的变种的势密度。一般来说，众所周知，这不是一个足以保证潜在密度的判据。Medvedev和Scanlon研究了可能出现的障碍，并提出了一个猜想，该猜想指出，如果在数域上定义的一个变体具有无限阶的自同构（或更一般地说是自同构），那么除非该变体映射到各种正维，其中自同构引起平凡自同构，否则它具有潜在密度。在与Ghioca和Tucker的合作中，我们证明了这一点。我想对某些类别的三倍研究这个问题。**理解这些情况下的势密度对于解决丢梵图问题（即当人们对相应的变种的自同构群有了一定的了解后）的某些方程的解有很大的帮助，并将增加我们对低维变种的算术性质的理解

This project deals with the problem of determining when a projective variety X defined over a number field has the so-called potential density property. That is, determining whether or not there exists a finite extension of our number field such that X has many (or, in more technical language, a Zariski dense) set of points over this larger number field. Although this problem is stated in geometric language, it enjoys a long history that predates modern algebraic geometry by hundreds of years. In its earliest form, one can see problems such as finding all Pythagorean triples or the problem of finding solutions to Pell's equation as variants of this problem where the varieties one is studying are in these cases curves defined over the rational numbers and one is attempting to find many different integer solutions.**The first interesting and not-fully-understood case is to understand potential density for surfaces and threefolds. Here there is already much known and there is a rough classification of surfaces due to Castelnuovo and Enriques that is of great help. This classification shows that up to some notion of equivalence, surfaces can be classified in terms of certain invariants. In terms of this classification, the potential density property for many of the classes of surfaces is completely understood. In some cases, potential density is only understood in terms of additional information. An example of this is recent work of Bogomolov and Tschinkel which shows that for a class of surfaces known as K3 surfaces one can prove the potential density property holds if the automorphism group is infinite. The automorphism group of a variety is in some sense giving a description of its symmetries and so saying that the variety has a large automorphism group is saying that it is not so rigid. ****My goal is to take the work of Bogomolov and Tschinkel as a starting point and study potential density of varieties with infinite automorphism groups. In general, it is known that this is not a sufficient criterion to ensure potential density. Medvedev and Scanlon have looked at the possible obstructions which can occur and have formulated a conjecture, which states that if a variety defined over a number field has an automorphism (or more generally an endomorphism) of infinite order then one has potential density unless the variety maps to a variety of positive dimension where the automorphism induces a trivial automorphism. In joint work with Ghioca and Tucker we proved this for surfaces. I would like to investigate this problem for certain classes of threefolds.**Understanding potential density in these cases would be of great benefit to those working in Diophantine problems of finding solutions to certain equations when one has some understanding of the automorphism group of the corresponding variety and would increase our understanding of the arithmetic properties of varieties of low dimension.**