Brauer–Manin obstruction
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| Brauer–Manin obstruction | |
|---|---|
| Name | Brauer–Manin obstruction |
| Field | Algebraic number theory |
| Introduced | 1970s |
Brauer–Manin obstruction is a reciprocity-based obstruction to the existence of rational points on algebraic varieties over global fields, formulated using the Brauer group and local adelic points. It explains some failures of the Hasse principle and refines descent obstructions by combining local invariants into a global pairing. The concept connects ideas from algebraic geometry, class field theory, and arithmetic geometry and has guided work by several mathematicians on rational points, Diophantine equations, and rationality problems.
Definition and basic examples
The obstruction was developed in the context of studies by Richard Brauer on central simple algebras, the Grothendieck school around Alexander Grothendieck, and later arithmetic applications by Yves Manin and others. Early examples include del Pezzo surfaces examined by researchers influenced by Jean-Pierre Serre and John Tate, where explicit counterexamples to the Hasse principle were constructed. Classical curves such as those arising from conic bundles and cubic surfaces studied by Federico Scorza-era school provided concrete instances; notable varieties exhibiting obstruction include certain Châtelet surfaces investigated by collaborators of Jean-Louis Colliot-Thélène and explicit constructions used by Bjorn Poonen and Martin Bright.
Brauer group and local-global principles
The Brauer group of a variety generalizes invariants studied by Richard Brauer and is related to cohomological constructions pioneered by Alexander Grothendieck and Jean-Louis Verdier. It interacts with local fields like completions studied by Kurt Hensel and global fields treated in the style of Emil Artin and John Tate. The obstruction refines classical local-global principles traced back to work of Harold Davenport and the Hasse school exemplified by Helmut Hasse and Ernst S. Selmer. Cohomological tools from the schools of Jean-Pierre Serre and Pierre Deligne provide the formalism linking the Brauer group with adelic points studied in the arithmetic of André Weil and the adelic techniques of James Tate.
The Brauer–Manin pairing
The key structure is a pairing between the Brauer group and the space of adelic points, echoing dualities developed by John Tate and reciprocity laws advanced by Emil Artin and Kurt Hensel. Manin's formulation uses local invariants at places studied in the spirit of Alexander Ostrowski and global reciprocity from the work of David Hilbert and the Hilbert symbol. The pairing detects incompatibilities among local points by summing invariants akin to reciprocity relations exploited by Claude Chevalley and later clarified by Jean-Pierre Serre's cohomological framework.
Applications to rational points and failures of Hasse principle
Applications include explanations for failures of the Hasse principle on curves and surfaces, a theme pursued by Jean-Louis Colliot-Thélène, Yves Hellegouarch-influenced researchers, and modern computational studies by Bjorn Poonen. Notable families where the obstruction is decisive include certain diagonal cubic surfaces investigated in the lineage of Ernst S. Selmer and Châtelet surfaces related to work by Jean-Marc Deshouillers-style analytic number theorists. The obstruction has been applied in the study of rational points on K3 surfaces within research traditions connected to André Weil and John Tate, and in rationality problems echoing classical questions addressed by Federico Scorza and later revived by Manjul Bhargava-style arithmetic investigations.
Computation and explicit methods
Explicit computation of the obstruction uses techniques developed in the computational arithmetic communities around William Stein and algorithmic number theory influenced by John Cremona. Methods employ cohomological calculations inspired by Alexander Grothendieck and descent techniques refined by Jean-Louis Colliot-Thélène and Bjorn Poonen. Practical implementations use local field data as in programs motivated by the numerical experiments of Noam Elkies and databases inspired by John Cremona and William Stein; these computations often adapt reciprocity computations influenced by David Hilbert's legacy.
Generalizations and related obstructions
Generalizations include étale-Brauer obstructions building on étale cohomology developed by Alexander Grothendieck and derived ideas from the work of Pierre Deligne and Jean-Pierre Serre. Related obstructions tie into descent theory as advanced by Yves Manin and Jean-Louis Colliot-Thélène and to obstructions studied in the context of the BSD conjecture lineage associated with Bryan Birch and Peter Swinnerton-Dyer. Connections to the arithmetic of automorphic forms evoke traditions from Robert Langlands and class field perspectives of Emil Artin.