On a conjecture of Agashe

The Birch and Swinnerton-Dyer (BSD) conjecture asserts that the size of the group of rational points of an elliptic curve, as well as several other invariants, are related to the behavior of an associated analytic object, the $L$-function of the curve. After discussing the BSD conjecture for elliptic curves over the rationals, we will focus on the analytic rank zero case and discuss a conjecture of Agashe, which is a consequence of the BSD. Finally, we will present a theorem that proves Agashe's conjecture.