Pfister forms

Pfister forms
Name	Pfister forms
Field	Algebra, Number Theory
Introduced	1960s
Discoverer	Albrecht Pfister

Pfister forms are a class of multiplicative quadratic forms introduced by Albrecht Pfister in the 1960s that play a central role in the theory of quadratic forms, the structure of the Witt ring, and related areas of algebra and arithmetic. They have been influential in the work of mathematicians such as Ernst Witt, Emil Artin, John Milnor, Alexander Grothendieck, and Jean-Pierre Serre, and they connect to topics studied by David Hilbert, Richard Dedekind, Helmut Hasse, and Kurt Gödel through their impact on field invariants and cohomological methods. Pfister forms serve as key examples and building blocks in investigations by researchers including Maxim Kontsevich, Vladimir Voevodsky, Johan de Jong, and Jacob T. Stafford.

Definition and basic properties

A Pfister form over a field F (of characteristic not 2, unless otherwise specified) is a special kind of n-fold tensor product of binary quadratic forms constructed from units in F. The basic 1-fold case is associated to a nonzero scalar a∈F× and yields the binary form <1, −a>, while an n-fold Pfister form is the tensor product of n such binary forms, producing a 2^n-dimensional quadratic form. Fundamental properties were established in work by Albrecht Pfister and later expanded by Eberhard Becker, Mikhail K. Kneser, Jean-Pierre Serre, and Alexander Merkurjev. Pfister forms are multiplicative in the sense that the values represented by a Pfister form are closed under multiplication, and they are either isotropic or anisotropic with tightly controlled behavior under field extensions studied by Oleg Viro and Richard Swan.

Construction and examples

Constructively, given a sequence (a1, a2, ..., an) of elements of F×, the corresponding n-fold Pfister form is the tensor product ⊗_{i=1}^n <1, −ai>. Examples include the 1-fold form <1, −a>, the 2-fold form <1, −a>⊗<1, −b> often written as <>, and higher-fold analogues <>. Classic instances arise over number fields studied by Carl Friedrich Gauss, Ernst Kummer, and David Hilbert: anisotropic Pfister forms occur over formally real fields investigated by Emil Artin and Helmut Hasse, while over finite fields results link to work of Richard Brauer and E. T. Parker. Explicit computational examples have been produced in studies by Manjul Bhargava, Jeroen Sijsling, and computer-aided research by groups at Princeton University, University of Cambridge, and École Normale Supérieure.

Algebraic theory and invariants

The algebraic study of Pfister forms interfaces with field cohomology, Galois cohomology, and Milnor K-theory. Pfister forms are connected to Milnor symbols in K_n^M(F)/2 via the norm residue isomorphism conjecture proven by Vladimir Voevodsky, Marc Rost, and Friedhelm I. K. Rost and clarified by work of Jean-Louis Colliot-Thélène and Alexander Merkurjev. Invariants such as the discriminant, the Clifford algebra, and higher cohomological invariants developed by John Milnor, Albrecht Pfister, and Jean-Pierre Serre classify Pfister forms up to isometry in many contexts. The behavior under scalar extension, decomposition into similar factors, and orthogonal sums links to investigations by Daniel Quillen, Hyman Bass, and Max Karoubi.

Relation to quadratic form theory and Witt ring

Pfister forms generate the fundamental filtration of the Witt ring W(F) of a field F introduced by Ernst Witt and studied by Max Zorn and R. Baeza. The ideal I^n(F) in W(F) has canonical generators represented by n-fold Pfister forms, and the structure of the successive quotients I^n/I^{n+1} relates to Milnor K-theory via the celebrated Milnor conjecture proven by Vladimir Voevodsky and Markus Rost. Results by T. Y. Lam, Pfister, and Karpenko demonstrate that Pfister forms determine important features of anisotropy, hyperbolicity, and splitting patterns in the Witt ring. The role of Pfister neighbors and similarity classes has been analyzed in work by Richard Elman, Nikolaus Karpenko, and Alexander Vishik.

Applications and notable results

Pfister forms have numerous applications across algebra, number theory, and algebraic geometry. They underpin examples and counterexamples in the study of isotropy of quadratic forms by Eberhard Becker, enable construction of fields with prescribed u-invariant as in work by David Hoffmann and Jean-Pierre Serre, and play a role in the classification of division algebras investigated by Richard Brauer and Amitsur. Key theorems include Pfister’s multiplicativity theorem, results on Pfister neighbors by Andrei Suslin and Ivan Panin, and connections to cohomological dimension studied by Serre and J.-P. Serre’s students. Pfister forms also appear in the arithmetic of quadratic forms over global fields researched by John Tate, Harold Hasse, and Dinakar Ramakrishnan.

Generalizations and related structures

Generalizations include generalized Pfister forms, multiplicative quadratic forms over rings explored by Max Karoubi and Daniel Quillen, and hermitian analogues over algebras with involution studied by A. A. Albert and M. Knebusch. Related structures encompass Clifford algebras, Azumaya algebras, and motivic cohomology objects investigated by Alexander Merkurjev, Voevodsky, and Spencer Bloch. Recent developments connect Pfister-type constructions to derived algebraic geometry themes pursued at Institut des Hautes Études Scientifiques and Mathematical Sciences Research Institute, and to automorphic forms and arithmetic duality studied by Robert Langlands and Pierre Deligne.

Category:Quadratic forms