The Brauer–Manin pairing, class field theory, and motivic homology

Abstract

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