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

TL;DR

This paper develops a residue-based approach to Grothendieck's trace map on arithmetic surfaces, extending reciprocity laws to all curves and connecting to adelic duality, thus advancing the understanding of arithmetic geometry.

Contribution

It introduces a residue-based description of Grothendieck's trace map and extends reciprocity laws to all curves on arithmetic surfaces, including points at infinity.

Findings

Residue-based description of the trace map

Extended reciprocity law to all curves including points at infinity

Applications to adelic duality for arithmetic surfaces

Abstract

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.