Grothendieck's trace map for arithmetic surfaces via residues and higher   adeles
  We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck's trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed. 
Matthew Morrow
math.AG math.NT