Brauer-Manin obstruction

Definition

The Brauer-Manin obstruction is a method used to understand the solvability of equations over global fields, particularly in the context of algebraic varieties. This concept connects local and global properties of varieties, illustrating how certain local conditions can prevent a global solution, even when solutions exist locally. It highlights the interplay between the Brauer group of a variety and rational points, contributing to broader themes such as the local-global principle, Hasse principle, and weak approximation.

5 Must Know Facts For Your Next Test

1. The Brauer-Manin obstruction often arises when trying to prove that a variety has no rational points despite having solutions in every local field.
2. This obstruction provides a way to construct counterexamples to the local-global principle by demonstrating cases where local conditions are satisfied but global conditions fail.
3. It is intimately connected with Galois cohomology, as the Brauer group can be viewed through the lens of cohomological methods.
4. The use of the Brauer-Manin obstruction has been particularly influential in studying rational points on higher-dimensional varieties, which can be more complex than curves or surfaces.
5. Certain classes of algebraic varieties, like K3 surfaces and certain conic bundles, have been shown to exhibit unique behaviors regarding rational points that are best understood through this obstruction.

Review Questions

• How does the Brauer-Manin obstruction provide insight into the local-global principle?
  • The Brauer-Manin obstruction reveals situations where local conditions are met—such as having solutions in all local fields—but global solutions fail due to specific restrictions imposed by the Brauer group. This contrasts with the local-global principle that would expect global solutions based on local data. Thus, it serves as a key tool for identifying exceptions and understanding why certain algebraic varieties may lack rational points despite local solvability.
• Discuss the relationship between the Brauer group and rational points on algebraic varieties, particularly in terms of examples where this connection is significant.
  • The Brauer group plays a vital role in determining the existence of rational points on algebraic varieties. For instance, in studying K3 surfaces or elliptic curves, one can compute the Brauer group to identify whether there are any obstructions to having rational points. If nontrivial elements in the Brauer group can be shown to prevent a variety from having rational solutions, this directly illustrates how local solutions may not extend globally, emphasizing the importance of these groups in Diophantine geometry.
• Evaluate how the Brauer-Manin obstruction interacts with weak approximation and what implications this has for understanding algebraic varieties.
  • The interaction between the Brauer-Manin obstruction and weak approximation reveals complex relationships between local and global behaviors of algebraic varieties. Weak approximation suggests that if a variety has rational points in all but finitely many places, it should have a global point. However, the Brauer-Manin obstruction can demonstrate that even when weak approximation holds, there may still be obstructions preventing global solutions. This nuance is crucial for understanding how various properties of varieties interlink and complicate our attempts to find rational solutions.

Related terms

Brauer Group: An algebraic structure that classifies central simple algebras over a field, playing a crucial role in the study of rational points on varieties.

Rational Points: Points on an algebraic variety whose coordinates are in the specified field, crucial for understanding the solutions to polynomial equations.

Local-Global Principle: A principle stating that if a property holds locally (at all completions of a field), it should also hold globally, though there are exceptions addressed by the Brauer-Manin obstruction.