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

TL;DR

This paper investigates how primes of good reduction influence the Brauer--Manin obstruction to weak approximation on varieties over number fields, providing conditions and examples, especially on K3 surfaces.

Contribution

It establishes necessary conditions on ramification indices for primes to be involved in the Brauer--Manin obstruction, supported by numerous examples on K3 surfaces.

Findings

Necessary conditions on ramification indices for primes of good reduction.

Identification of primes involved in Brauer--Manin obstruction.

Examples of transcendental Brauer--Manin obstruction on K3 surfaces.

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.