Grothendieck's trace map for arithmetic surfaces via residues and higher adeles

Details Grothendieck's trace map for arithmetic surfaces via residues and higher adeles Grothendieck's trace map for arithmetic surfaces via residues and higher adeles

2011-01-10

arxiv.org/abs/1101.1883v2

Mathematics

Algebraic Geometry

Update to previous version of paper: most important change is a proof of the horizontal reciprocity law on arithmetic surfaces

Scientific paper

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.

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