# 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.

## Key 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.