Brauer-Manin obstructions requiring arbitrarily many Brauer classes

TL;DR

This paper demonstrates that for certain algebraic varieties over global fields, the minimal Brauer group obstructions to rational points can be arbitrarily complex, needing many generators.

Contribution

It proves that the finite Brauer subgroup obstructing rational points can require arbitrarily many generators, highlighting the complexity of Brauer-Manin obstructions.

Findings

Brauer groups can require arbitrarily many generators for obstructions.

Finite Brauer subgroups can be arbitrarily large.

Obstructions are not uniformly bounded in complexity.

Abstract

On a projective variety defined over a global field, any Brauer--Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.