Elliptic curves and the Birch and Swinnerton-Dyer conjecture

We will discuss certain arithmetic aspects of elliptic curves, with a view towards the Birch and Swinnerton-Dyer conjecture. We will give more details of some of the objects involved compared to earlier talks given here. In particular, we will discuss why the group of rational points is finitely generated, a fact that is needed to even state the first part of the conjecture. In the process, we will naturally come across the Shafarevich-Tate group, which is one of the objects in the second part of the conjecture.