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

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 conditions on the ramification index of primes involved in the Brauer-Manin obstruction.To support the results, we give many examples of transcendental nature on K3 surfaces.We will address the following question.Question 1.2.Assume Pic(V ) to be torsion-free and finitely generated.Which primes can play a role in the Brauer-Manin obstruction to weak approximation on V ?This question is inspired by one asked by Swinnerton-Dyer [Colliot-Thélène and Skorobogatov 2013, Question 1].He asked whether only primes of bad reduction and archimedean places can play a role in MSC2020: primary 14F22, 14G05, 14G12, 14G17, 14G20; secondary 14F30.