The Brauer-Manin obstruction to the local-global principle for the embedding problem

TL;DR

This paper explores the Brauer-Manin obstruction in the context of embedding problems over global fields, establishing key structural results and characterizations related to weak approximation and duality.

Contribution

It introduces an analogue of the Brauer-Manin obstruction for embedding problems, proving its sufficiency for weak approximation with abelian kernels and providing new insights into Tate duality and cup products.

Findings

Brauer-Manin obstruction is the only obstruction to weak approximation for abelian kernel embedding problems.

Provides a new, elegant description of Tate duality pairing.

Proves a new theorem on the cup product.

Abstract

We study an analogue of the Brauer-Manin obstruction to the local-global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic) Brauer-Manin obstruction is the only one to weak approximation when the embedding problem has abelian kernel. As a part of our investigations we also give a new, elegant description of the Tate duality pairing and prove a new theorem on the cup product.