Brauer–Manin obstruction

Hasse theorem states that a quadratic form has a rational point over a number field if and only if this quadratic form is isotropic over all completions of the given number field. However, this principle does not hold in the general case. Yuri Ivanovich Manin invented the recipe to search for obstruction to Hasse principle. We are going to discuss this recipe and try to apply it for an elliptic curve and some del Pezzo surfaces.