BrauerManin

BrauerManin is a concept in algebraic geometry that arises from the study of algebraic groups and their representations over fields of positive characteristic. Specifically, it relates to the structure of certain algebraic groups and their behavior when the underlying field has a characteristic $p > 0$.

The term "BrauerManin" is often associated with the Brauer group of a field, which classifies central simple algebras over that field. In the context of algebraic groups, the Brauer group can be used to study obstructions to the existence of rational points on these groups. The Manin-Gromov method, developed by Yuri Manin and others, provides a framework for analyzing these obstructions, particularly for varieties like Severi-Brauer varieties.

The BrauerManin obstruction is a phenomenon where the non-vanishing of certain elements in the Brauer group of a field implies the non-existence of rational points on an algebraic variety, even if local conditions (like the Hasse principle) are satisfied. This obstruction is particularly relevant for homogeneous spaces of algebraic groups.

Research in this area often involves understanding the relationship between the Picard group of a variety, its Brauer group, and the structure of the underlying algebraic group. It is a sophisticated area of mathematics with applications in number theory and arithmetic geometry.