Applications of arithmetic dynamics to potential density

这个项目处理的问题，确定当一个射影簇X定义在数域上有所谓的潜在的密度性质。 也就是说，确定是否存在我们的数域的有限扩张，使得X在这个更大的数域上有许多（或者用更专业的语言说，一个Zebraki稠密）点集。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几百年。 在其最早的形式中，人们可以看到诸如寻找所有毕达哥拉斯三元组或寻找佩尔方程的解的问题作为这个问题的变体，其中正在研究的变体在这些情况下是定义在有理数上的曲线，并且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曲面和三重的势密度。 在这里已经有很多已知的，有一个粗略的分类表面由于卡斯泰尔诺沃和恩里克斯是很大的帮助。 这种分类表明，直到某种等价概念，曲面可以根据某些不变量进行分类。 根据这种分类，许多类别的表面的潜在密度属性是完全理解的。 在某些情况下，潜在的密度只能从附加信息的角度来理解。 这方面的一个例子是最近的工作Bogomolov和Tschinkel这表明，一类表面被称为K3表面可以证明潜在的密度性质举行，如果自同构群是无限的。 一个簇的自同构群在某种意义上是对它的对称性的描述，所以说这个簇有一个大的自同构群就是说它不是那么严格。* 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簇的潜在密度。 一般来说，已知这不是确保潜在密度的充分标准。梅德韦杰夫和斯坎伦研究了可能发生的障碍，并提出了一个猜想，即如果定义在数域上的一个簇有一个无限阶的自同构（或更一般的自同构），那么它就有潜在的密度，除非这个簇映射到一个正向的维数，在这个维数上，自同构诱导了一个平凡的自同构。在联合工作与吉奥卡和塔克我们证明了这一点的表面。 我想研究这个问题的某些类的三倍。**了解这些情形下的势密度，对于那些研究丢番图问题的人来说是非常有益的，因为他们需要对相应簇的自同构群有一定的了解，并且会增加我们对低维簇的算术性质的理解。

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.**