On the Brauer group of an arithmetic model of a hyperkähler variety over a number field

Abstract

We prove Artin's conjecture on the finiteness of the Brauer group for an arithmetic model of a hyperkähler variety over a number field provided that . We show that the Brauer group of an arithmetic model of a simply connected Calabi–Yau variety over a number field is finite. We also prove that if the -adic Tate conjecture on divisors holds for a certain smooth projective variety over a field of arbitrary characteristic , then the group is finite independently of the semisimplicity of the continuous -adic representation of the Galois group on the space .