On the Brauer group of diagonal quartic surfaces

TL;DR

This paper provides a simple criterion to determine when the Brauer group of a diagonal quartic surface over Q is algebraic, and offers bounds on the quotient of the Brauer group by the base field's Brauer group.

Contribution

It introduces a new sufficient condition for the algebraicity of the Brauer group of diagonal quartic surfaces and establishes bounds on related quotient groups.

Findings

A simple sufficient condition for algebraic Brauer groups of diagonal quartic surfaces.

An upper bound for the order of the Brauer group quotient.

Utilization of Fermat quartic and Kummer surface isomorphism in proofs.

Abstract

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Masumi Mizukami.