Weak approximation and non-abelian fundamental groups

We introduce a new obstruction to weak approximation related to non-abelian coverings of a proper and smooth variety X deened over a number eld k. It provides some counterexamples to weak approximation which are not accounted for by the Manin obstruction, for example bielliptic surfaces. 0. Introduction Let X be a smooth and proper algebraic variety over a number eld k and k be the set of places of k. Recall (cf. CT/San87], Sko98]) that X is a counterexample to the Hasse principle if the set X(A k) := Q v2 k X(k v) of adelic points of X is non-empty, but the set X(k) of k-rational points of X is empty. A k-variety X (such that X(k) 6 = ;) satisses weak approximation if X(k) is dense in X(A k) (equipped with the product of the v-adic topologies). It satisses weak approximation outside S (where S is a nite set of places of k) if X(k) is dense in X(A S k) := Q v6 2S X(k v). In his talk at the ICM in 1970 ((Ma70]), Manin deened an obstruction to the Hasse principle, the so-called Brauer-Manin obstruction. A similar obstruction to weak approximation was later deened by Colliot-Th el ene and Sansuc (cf. CT/San87], section 3). Those obstructions are related to the Brauer group BrX of X (we shall recall their precise deenitions in 1.3.). For a long time, all known counterexamples to the Hasse principle and to weak 1