Diagonal quartic surfaces and transcendental elements of the Brauer group

TL;DR

This paper studies the Brauer group of diagonal quartic surfaces, providing explicit algebraic representatives, conditions for algebraicity over number fields, and identifying transcendental obstructions to weak approximation.

Contribution

It constructs explicit central simple algebras representing the 2-torsion of the Brauer group and analyzes algebraic versus transcendental parts over number fields.

Findings

Explicit construction of central simple algebras for the 2-torsion Brauer group

Conditions under which the 2-primary Brauer group is algebraic

Identification of a transcendental obstruction to weak approximation

Abstract

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.