Brauer-Manin obstructions on degree 2 K3 surfaces

TL;DR

This paper investigates the Brauer-Manin obstruction on degree 2 K3 surfaces over rationals, identifying infinite families of counterexamples to the Hasse principle explained by algebraic Brauer groups.

Contribution

It explicitly computes the geometric Picard group of these K3 surfaces and identifies two types of algebraic Brauer-Manin obstructions, answering a question by Ieronymou and Skorobogatov.

Findings

Found explicit generators for the geometric Picard group.

Identified two infinite families of counterexamples to the Hasse principle.

Demonstrated obstructions from quaternion algebra and 3-torsion elements.

Abstract

We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.