• Heegner points
• global class field theory
• Birch and Swinnerton-Dyer conjecture
• Kolyvagin's theorem
• ring class fields

In this thesis we present the main theorems of global class field theory stated in terms of ideals and we describe an application of class field theory to the study of the group of rational points of an elliptic curve.

By using the properties of Heegner points and results in Galois cohomology, in the late eighties Kolyvagin proved that, given E an elliptic curve defined over the rationals and K a quadratic imaginary field, under suitable hypothesis on the field K and on the Heegner points defined over the ring class fields of K, the rank of E(K) is one. This theorem has been vastly studied and the techniques involved in its proof have been subsequently used in a variety of contexts.
Moreover, this theorem, combined with other results, leads to the proof of part of the Birch and Swinnerton-Dyer conjecture.

In the thesis, after proving the theorems of global class field theory in the language of ideals, we give the definition of ring class fields and describe their explicit construction by means of modular functions. Then, following mainly Darmon's "Rational points on modular elliptic curves", we describe the proof of a weaker result than Kolyvagin's theorem, which, avoiding some technical difficulties, still implies the most part of Kolyvagin's result and presents the main points of the techniques involved. Finally, we describe how Kolyvagin's theorem can be used, together with other results, to prove part of the Birch and Swinnerton-Dyer conjecture.