Non-abelian descent and the arithmetic of Enriques surfaces

Abstract

The Brauer-Manin obstruction to the Hasse principle and weak approximation provides a fruitful general approach to rational points on varieties over number fields. A fundamental problem here can be stated as follows: is it possible to describe in purely geometric terms the class of smooth projective varieties for which the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation? In recent examples where the Brauer-Manin obstruction is not the only one (see [1, 7, 14]), the key role is played by etale Galois coverings with a non-abelian Galois group. This has left open the question whether similar examples exist for varieties with an abelian geometric fundamental group. The case of principal homogeneous spaces of abelian varieties and that of rational surfaces (which are geometrically simply connected), where the Brauer-Manin obstruction is expected to be the only one, might seem to suggest that as long as the geometric fundamental group is abelian, the Brauer-Manin obstruction should still be the only one. The Manin obstruction was linked to the classical abelian descent by ColliotThelene and Sansuc [2]. In [8], the authors introduced the non-abelian descent as a new tool for studying rational points. The present paper enriches the non-abelian theory with a general method for constructing non-abelian torsors, and then applies it to an example which answers the above question in the negative.