Relative K-groups and class field theory for arithmetic surfaces

Here CH0(X) denotes the Chow group of zero-cycles on X modulo rational equivalence and π̃ 1 (X) ab is the modified abelianized étale fundamental group, which classifies finite abelian étale coverings of X in which every real point splits completely (if X is defined over a totally imaginary number field, then this is just the usual abelianized étale fundamental group). By Lang [La], the homomorphism recX has a dense image and (after S. Bloch [Bl] solved special cases) K. Kato and S. Saito [K-S1], [K-S2], proved the following theorem which is usually cited as “unramified class field theory”.