Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

TL;DR

This paper investigates how odd torsion elements of the Brauer group influence the arithmetic of diagonal quartic surfaces over number fields, showing conditions under which they do not obstruct weak approximation.

Contribution

It introduces new criteria for when odd torsion Brauer elements do not hinder weak approximation on diagonal quartic surfaces over arbitrary number fields.

Findings

Odd torsion Brauer elements induce constant evaluation maps over local fields with coprime order and residue characteristic.

A mild, checkable condition ensures odd torsion does not obstruct weak approximation over number fields.

Provides a systematic method to construct K3 surfaces over  with specific Brauer group properties.

Abstract

We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also note a systematic way to produce $K3$ surfaces over $\Q_2$ with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.