Embeddings of maximal tori in classical groups and explicit Brauer-Manin obstruction

TL;DR

This paper provides necessary and sufficient conditions for embedding maximal tori into classical groups over global fields, analyzing the Hasse principle through a Brauer-Manin obstruction framework.

Contribution

It offers a complete characterization of the embedding problem and the Hasse principle for classical groups over global fields, extending previous work with explicit criteria.

Findings

Conditions for the existence of embeddings are fully characterized.

The Hasse principle holds precisely when a Brauer-Manin obstruction vanishes.

The work connects embedding problems with Brauer-Manin obstructions in a new way.

Abstract

Embeddings of maximal tori into classical groups over global fields of characteristic not 2 are the subject matter of several recent papers, with special attention to the Hasse principle. The present paper gives necessary and sufficient conditions for this embedding problem, and in particular for the Hasse principle to hold. Using work of Borovoi, this is interpreted as a Brauer-Manin type obstruction.