Applications of arithmetic dynamics to potential density

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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.