Brauer-Manin pairing, class field theory and motivic homology

TL;DR

This paper explores the relationship between the Brauer-Manin pairing, class field theory, and motivic homology for smooth varieties over p-adic fields, extending known results to non-proper cases.

Contribution

It generalizes the connection between Brauer groups, fundamental groups, and Chow groups to non-proper varieties using motivic homology and tame class groups.

Findings

Extended the Brauer-Manin pairing to non-proper varieties.

Connected motivic homology with class field theory concepts.

Provided examples illustrating the generalized relations.

Abstract

For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.