Applications of arithmetic dynamics to potential density

本课题研究了定义在数域上的射影簇X何时具有所谓的势密度性质的问题。也就是说，确定我们的数域是否存在有限扩展，使得X在这个较大的数域上有许多(或者，用更专业的语言来说，是Zariski稠密的)点集。虽然这个问题是用几何语言来描述的，但它的历史比现代代数几何早了几百年。在其最早的形式中，人们可以将寻找所有毕达哥拉斯三元组或寻找佩尔方程的解的问题视为该问题的变体，在这些情况下，人们研究的变量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。**第一种有趣且尚未完全理解的情况是理解曲面和三重数的势密度。由于Castelnuovo和Enrique，这里已经有了很多已知的表面，并且有一个粗略的分类，这是非常有帮助的。这种分类表明，在某种等价的概念下，曲面可以根据某些不变量来分类。根据这种分类，许多类曲面的势密度性质是完全理解的。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，该工作表明，对于一类被称为K3曲面的曲面，如果自同构群是无限的，则可以证明势密度性质成立。一个簇的自同构群在某种意义上是对它的对称性的一种描述，所以说这个簇有一个大的自同构群就是说它不那么严格。*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位势密度。一般来说，众所周知，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簇具有无穷阶的自同构(或更一般地说，自同态)，则一个簇具有势密度，除非该簇映射到自同构导致平凡自同构的各种正维度。在与Ghioca和Tucker的合作中，我们证明了曲面的这一点。我想研究某些三重数类的这个问题。**了解这些情况下的势密度将对那些研究丢番图问题的人有很大的帮助，当一个人对相应簇的自同构群有一些了解时，就可以找到某些方程的解，并将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.**