The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.

In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture.