On the Brauer group of an arithmetic scheme

Abstract

For an Enriques surface over a number field with a -rational point we prove that the -component of is finite if and only if . For a regular projective smooth variety satisfying the Tate conjecture for divisors over a number field, we find a simple criterion for the finiteness of the -component of . Moreover, for an arithmetic model of we prove a variant of Artin's conjecture on the finiteness of the Brauer group of . Applications to the finiteness of the -components of Shafarevich-Tate groups are given.