Transcendental obstructions to weak approximation on general K3 surfaces

TL;DR

This paper constructs a specific K3 surface over rationals with Picard rank one that exhibits a transcendental Brauer-Manin obstruction to weak approximation, linking K3 surfaces and cubic fourfolds.

Contribution

It provides an explicit example of a K3 surface with a transcendental obstruction to weak approximation, highlighting new connections between K3 surfaces and cubic fourfolds.

Findings

Constructed an explicit K3 surface with geometric Picard rank one over rationals.

Demonstrated a transcendental Brauer-Manin obstruction to weak approximation on this surface.

Linked properties of polarized K3 surfaces with special Brauer classes to cubic fourfolds.

Abstract

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.