ar X iv : 0 80 4 . 34 19 v 2 [ m at h . N T ] 1 7 M ar 2 00 9 Covering data and higher dimensional global class field theory Moritz Kerz and

For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism ρ X : C X → π ab 1 (X), which is surjective and whose kernel is the connected component of the identity. The (topological) group C X is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend. The aim of global class field theory is the description of abelian extensions of arithmetic schemes (i.e. regular schemes X of finite type over Spec(Z)) in terms of arithmetic invariants attached to X. The solution of this problem in the case dim X = 1 was one of the major achievements of number theory in the first part of the previous century. In the 1980s, mainly due to K. Kato and S. Saito [8], a generalization to higher dimensional schemes has been found. The description of the abelian extensions is given in terms of a generalized idèle class group, whose rather involved definition is based on Milnor K-sheaves. In the course of the last years, G. Wiesend developed a new approach to higher dimensional class field theory which only uses data attached to points and curves on the scheme. The central and new idea was to consider data which describe not necessarily abelian Galois coverings of all curves on the scheme, together with some compatibility condition. Then one investigates the question whether these data are given by a single Galois covering of the scheme. The essential advantage of this nonabelian approach is that one can use the topological finite generation of the tame fundamental groups of smooth curves over separably closed fields as an additional input. The restriction to abelian coverings is made at a later stage. One obtains an explicitly given class group C X together with a reciprocity ho-momorphism ρ X : C X → π ab 1 (X) to the abelianized fundamental group, which has similar properties like the classical reciprocity homomorphism of one-dimensional class field theory. As a result of the method, the full abelian fundamental group can be described only if X is flat over Spec(Z) and for varieties over finite fields which are proper over a curve. For a general variety over a finite field, the method only yields a …