On Potential Density of rational Points on Abelian Varieties

One says that an algebraic variety V defined over a field K has po-tential density of rational points if after some finite extension of fields L/K the set V (L) is a Zariski dense set. Here we will be interested in the case where K is a number field. In the case of curves, there is a complete classification of potential density based on the genus of the given curve. In this project, our aim is to discuss what is known in the case of surfaces and what is expected for varieties in general. In particular, we are going to show that potential density holds for abelian varieties.