arithmetic duality theorems

arithmetic duality theorems
Name	Arithmetic duality theorems
Field	Number theory, Algebraic geometry
Notable persons	John Tate, Georges Poitou, Jean-Pierre Serre, Alexander Grothendieck, Jean-Louis Verdier, Jean-Pierre Serre, Barry Mazur, Pierre Deligne, Alexander Beilinson, Vladimir Voevodsky
Institutions	Harvard University, Institut des Hautes Études Scientifiques, École Normale Supérieure, University of Cambridge, Princeton University
First published	1960s–1970s

arithmetic duality theorems are a collection of results in number theory and algebraic geometry that relate cohomology groups of Galois modules, abelian varieties, and arithmetic schemes via pairings and exact sequences. They connect local and global phenomena through dualities originally formulated by John Tate and Georges Poitou, developed using tools introduced by Jean-Pierre Serre, Alexander Grothendieck, and Jean-Louis Verdier. These theorems underpin structural results about class field theory, the arithmetic of elliptic curves, and the study of Selmer and Shafarevich–Tate groups encountered in work by Barry Mazur, Pierre Deligne, and others.

Overview and Motivation

The motivation for arithmetic duality arises from classical reciprocity laws in Kronecker's and Hilbert's formulations and culminates in modern cohomological formulations influenced by Alexander Grothendieck's development of étale cohomology and duality theories used by Jean-Pierre Serre and Jean-Louis Verdier. Early goals included clarifying the relation between idele class groups studied in Richard Dedekind's and Emil Artin's frameworks and cohomology groups used by John Tate and Georges Poitou to formulate local-global principles that later informed work of André Weil and Alexander Grothendieck on duality for schemes.

Classical Duality Theorems (Tate, Poitou–Tate)

The classical theorems are epitomized by John Tate's local duality for finite Galois modules and the global Poitou–Tate exact sequence due to Georges Poitou and refinements by John Tate. Tate duality provides perfect pairings between Galois cohomology groups of a finite module and its Cartier dual, echoing principles in Hilbert's reciprocity and connecting with Emil Artin's reciprocity map. The Poitou–Tate sequence links global Galois cohomology, local cohomology at places associated to Chevalley's idèle theory, and ramifications controlled by inertia groups prominent in work of Claude Chevalley and Helmut Hasse.

Cohomological Formalism and Local Duality

Formulation relies on derived functors and Verdier duality introduced by Jean-Louis Verdier within derived categorys shaped by Alexander Grothendieck's SGA seminars, building on foundational insights of Jean-Pierre Serre. Local duality theorems identify a trace pairing between étale cohomology with compact support and ordinary cohomology for local fields studied by Hermann Weyl and Emil Artin's successors, producing perfect pairings that are central in the analysis of torsion phenomena in Galois representations arising in Pierre Deligne's work on the Weil conjectures.

Duality for Abelian Varieties and Galois Modules

Duality statements for abelian varieties—notably for elliptic curves and Jacobians studied by André Weil and Jacques Hadamard—relate the Galois cohomology of torsion points to dual isogeny modules via the Cassels pairing and the Weil pairing, exploited by Barry Mazur and John Tate in arithmetic investigations. For finite Galois modules, Cartier duality and Pontryagin duality interplay with Tate local pairings; these ideas have been applied in the study of Selmer groups and the Shafarevich–Tate group in contexts explored by Gerd Faltings and Andrew Wiles.

Global-to-Local Exact Sequences and Applications

Poitou–Tate global-to-local exact sequences furnish long exact sequences that compare global cohomology groups to products of local cohomology groups over completions at places appearing in Chevalley's idèle framework and used in class field theory by Emil Artin and Helmut Hasse. Applications include finiteness results and obstruction theories for principal homogeneous spaces studied in the context of the Birch and Swinnerton-Dyer conjecture pursued by John Tate and Bryan Birch, descent methods in the style of Manjul Bhargava and Joseph Oesterlé, and reciprocity constraints used by Ken Ribet and Andrew Wiles in modularity arguments.

Generalizations and Modern Approaches (Étale, Poitou–Tate refinements)

Modern treatments generalize classical duality using the framework of étale cohomology advanced by Alexander Grothendieck, derived categories and perverse sheaves developed by Pierre Deligne and Jean-Louis Verdier, and motivic perspectives from Alexander Beilinson and Vladimir Voevodsky. Refinements of Poitou–Tate accommodate completed cohomology and p-adic Hodge theoretic inputs as in work by Kazuya Kato, Jean-Marc Fontaine, and Richard Taylor. These approaches inform contemporary research on Iwasawa theory pursued by Ralph Greenberg and Kenkichi Iwasawa, questions about Bloch–Kato conjectures examined by Spencer Bloch and Kazuya Kato, and the analysis of non-commutative generalizations studied by Coates and John Coates.

Category:Number theory