Let be a regular arithmetic surface. Assume that for all irreducible curves there are given open normal subgroups of 1(C), which fulfill a compatibility condition at all closed points x . We then show that these data uniquely determine a normal subgroup of 1(). This is used to construct abelian class field theory for arithmetic surfaces using only K0 and K1 groups of local and global fields.Mathematics Subject Classification (2000): 14G40, 11R37, 14H25