Quantitative arithmetic of diagonal degree $2$ K3 surfaces

TL;DR

This paper investigates rational points on a family of diagonal degree 2 K3 surfaces over the rationals, analyzing the Brauer group and Brauer-Manin obstructions, revealing that odd torsion explains certain failures of the Hasse principle.

Contribution

It provides the first detailed analysis of Brauer groups for this family, showing that the Brauer group is almost always trivial and quantifying the surfaces with Brauer-Manin obstructions.

Findings

Brauer group is almost always trivial for these surfaces.

Quantifies the frequency of Brauer-Manin obstructions among the family.

Shows that odd order torsion explains certain failures of the Hasse principle.

Abstract

In this paper we study the existence of rational points for the family of K3 surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost always trivial, and find the exact order of magnitude of surfaces for which there is a Brauer-Manin obstruction to the Hasse principle. Our results show definitively that K3 surfaces can have a Brauer-Manin obstruction to the Hasse principle that is only explained by odd order torsion.