Georgia Institute of Technology College of Sciences

The rational solutions to the equation describing an elliptic curve form a finitely generated abelian group, also known as the Mordell-Weil group. Detemining the rank of this group is one of the great unsolved problems in mathematics. The Shafarevich-Tate group of an elliptic curve is an important invariant whose conjectural finiteness can often be used to determine the generators of the Mordell-Weil group. In this talk, we first introduce the definition of the Shafarevich-Tate group. We then discuss the theory of visibility, initiated by Mazur, by means of which non-trivial elements of the Shafarevich-Tate group of an elliptic curve an be 'visualized' as rational points on an ambient curve. Finally, we explain how this theory can be used to give theoretical evidence for the celebrated Birch and Swinnerton-Dyer Conjecture.