Good reduction of the Brauer-Manin obstruction

TL;DR

This paper demonstrates that for certain smooth projective varieties over number fields, the Brauer-Manin obstruction can be effectively described by considering only specific places, simplifying the analysis of rational points.

Contribution

It establishes a reduction theorem showing that only archimedean places, bad reduction primes, and primes dividing the transcendental Brauer group influence the Brauer-Manin set.

Findings

Only archimedean places affect the Brauer-Manin obstruction.

Primes of bad reduction are relevant for the obstruction.

Primes dividing the transcendental Brauer group are significant.

Abstract

For a smooth and projective variety over a number field with torsion free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set.