In this chapter we discuss the Hasse principle and various approximation properties for varieties defined over number fields, as well as known obstructions to them.

The short survey of known results on the Hasse principle and weak approximation in the first section is just an introduction to the main object of interest here: obstructions and their interrelations. Fortunately, this subject is covered in a few excellent survey articles [Sansuc 82], [MT], [Sansuc 87], [C87], [C92], [SD96], [C98]. The Manin obstruction to the Hasse principle, and its various Ramifications, are defined in the second section.

In the last section we discuss the obstructions to the Hasse principle and weak approximation obtained via descent with torsors under (possibly, noncommutative) algebraic groups. Examples (Chapter 8) show that these obstructions can be finer than the Manin obstruction. The classical theory of descent on elliptic curves and their principal homogeneous spaces worked with torsors under finite abelian (algebraic) groups. This is covered by the Manin obstruction. To go beyond it one applies the same idea in the nonabelian setting.

The Hasse principle, weak and strong approximation

A class of geometrically integral varieties over a number field k satisfies the Hasse principle if for every variety in this class the condition X(ku) ≠ ∅ for all places v of k implies X(k) ≠ ∅. Usually we shall speak of the Hasse principle for smooth varieties.