On the Birch and Swinnerton-Dyer Conjecture

Birch and Swinnerton-Dyer Conjecture (BSD-Conjecture) is one of the most popular problems of Mathematics in recent years and it became a more interesting problem with Clay Mathematics Institute's $1 million prize for its solution. Numerical data support this conjecture's validity. The conjecture, which connects an algebraic object and an analytic object in elliptic curves, consists of two parts. The first part is related to rank and the second part claims the validity of a formula related to elliptical curves. The aim of this four-parted study is to introduce the BSD-conjecture. The first part is spared for the properties of elliptic curves. In the second part, the part related to the rank of BSD-Conjecture is stated and current results in the literature review are discussed. In the third part, Tate-Shafarevich groups of elliptic curves are introduced and the formula which composes the second part of BSD- Conjectures is given. In the fourth and the last section, some examples that support the conjecture are discussed. The study is conducted as a compilation.