Contents: Algebraic Methods: Finite fields. p-adic fields. Hilbert symbol. Quadratic forms over Qp, and over Q. Integral quadratic forms with discriminant -1.- Analytic Methods: The theorem on arithmetic progressions. Modular forms.- Bibliography.- Indices. This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapter ...