The role of primes of good reduction in the Brauer--Manin obstruction

Abstract: We discuss the role of primes of good reduction in the existence of the Brauer--Manin obstruction to weak approximation for varieties defined over number fields. Following Bright and Newton, we give some necessaries conditions on the ramification index that the prime ideal of the number field should satisfy in order to be involved in the Brauer--Manin obstruction to weak approximation. To support the results, many examples of transcendental Brauer--Manin obstruction to weak approximation on K3 surfaces are given.