Two-dimensional local-global class field theory in positive characteristic

Using the higher tame symbol and Kawada and Satake’s Witt vector method, A.N. Parshin developed class field theory for positive characteristic higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases. We prove reciprocity laws for these symbols using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol. We then extend this method of defining a reciprocity map to the case of positive characteristic local- global fields associated to points and curves on an algebraic surface over a finite field