Elliptic curves with complex multiplication and applications to class field theory

One of the aims of algebraic number theory is to describe the field of algebraic numbers and the extensions of number fields. This problem appears as the 12° of the 23 Hilbert's problems, and is essentially an extension of the Kronecker-Weber theorem, from the field of rational numbers to a generic number field. Although the problem is still open, the particular case of quadratic imaginary fields is completely understood, thanks to the theory of elliptic curves with complex multiplication. The purpose of this dissertation is to introduce some definitions and properties of elliptic curves (in Chapter 1), of the complex multiplication on them (in Chapter 2), of the class field theory (in Chapter 3) and then to give a characterization of the maximal abelian extension and then of any abelian extension of quadratic imaginary fields, with some other interesting properties about elliptic curves with complex multiplication (in Chapter 4).