We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for surfaces over a perfect field. In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide.

Much of the work in this paper was done while I was the recipient of the Cecil King Travel prize, through the London Mathematical Society; I would like to thank the Cecil King Foundation for their generosity.