Kolyvagin's work on modular elliptic curves

The main achievements towards the proof of the Birch and Swinnerton-Dyer conjecture have been attained in the case that the considered elliptic curve has rank equal to 0 or 1, thanks to the work of Kolyvagin, Gross, Zagier, Rubin and others. In this thesis we are going to discuss the ideas and the tools that were used to obtain those results, and we present the proof one of these theorems. After having reviewed the basics of group cohomology and the definitions of the Selmer group and the Shafarevich-Tate group, we introduce some concepts of class field theory: more precisely, the ring class fields of an imaginary quadratic field. These fields will be fundamental in the construction of a collection of points on E, called Heegner points, rational over these same fields. Using these points we define the Kolyvagin classes, distinguished elements of the first cohomology group of the p-torsion points of E that will play an essential role throughout all the proofs.