An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction

TL;DR

This paper constructs a specific K3 surface over the rationals demonstrating a Brauer-Manin obstruction to weak approximation solely at the prime 2, which has good reduction, illustrating a nuanced arithmetic phenomenon.

Contribution

It provides an explicit example of a K3 surface with good reduction at 2 where weak approximation fails only at that prime, highlighting a new case of Brauer-Manin obstruction.

Findings

Weak approximation is obstructed only at prime 2.

The constructed K3 surface has good reduction at 2.

This example illustrates a specific Brauer-Manin obstruction scenario.

Abstract

Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.