Quadratic forms

Quadratic forms
Name	Quadratic forms
Field	Mathematics

Quadratic forms are homogeneous polynomial expressions of degree two in several variables, central to algebra, number theory, and geometry. They connect classical problems studied by Carl Friedrich Gauss, Joseph-Louis Lagrange, and Johann Carl Friedrich Gauß with modern developments involving Emmy Noether, David Hilbert, and John von Neumann. Quadratic forms influence the work of researchers at institutions such as École Normale Supérieure, Institute for Advanced Study, and Max Planck Society while interacting with problems posed by organizations like the American Mathematical Society and events such as the International Congress of Mathematicians.

Definition and Basic Properties

A quadratic form on a vector space over a field F or a module over a ring R is a function Q: V → F (or R) given by a homogeneous degree-two polynomial; this formalism appears in texts by Isaac Newton and in the classification efforts of Euclid through later expositions by Leonhard Euler and Augustin-Louis Cauchy. For a finite-dimensional space with basis vectors the form corresponds to a symmetric matrix A, linking to linear operators studied by Arthur Cayley and Camille Jordan. Basic properties include nondegeneracy, rank, and signature; these notions were refined in work by Hermann Minkowski, Emil Artin, and André Weil. Change of basis by matrices in Élie Cartan's matrix groups preserves equivalence classes; such transformations are central to the study by Évariste Galois and exploited in classification theorems akin to those of David Hilbert.

Examples and Canonical Forms

Standard examples include the Euclidean norm x_1^2 + ... + x_n^2 linked historically to Pythagoras and later formalized by Carl Gustav Jacobi. The hyperbolic form x_1 x_2 arises in investigations by Sophus Lie and in models used by Hermann Weyl. Over the reals, Sylvester's law of inertia, proved in contexts connected to James Sylvester and Arthur Cayley, yields the signature canonical form; this result is used by Bernhard Riemann in curvature theory and by Henri Poincaré in topological classifications. Over finite fields, canonical diagonalizations relate to work by Émile Borel and applications considered by Claude Shannon in coding theory contexts. In the p-adic setting the classification reflects contributions of Kurt Hensel and John Tate; explicit representatives and normal forms feature in studies by Igor Shafarevich and Jean-Pierre Serre.

Classification over Fields and Rings

Classification over the real numbers uses signatures tied to results by David Hilbert and forms analyzed by Hermann Minkowski; over algebraic number fields the theory incorporates local-global principles developed by Helmut Hasse and consolidated by Erich Hecke. Over p-adic fields and completions the Hasse–Minkowski theorem, influenced by correspondence between Helmut Hasse and Hermann Minkowski, yields equivalence criteria used by John Milnor and Kentaro Nagao. Over rings such as the integers, integral quadratic forms were central to work by Joseph-Louis Lagrange (four-square theorem) and later by G. L. Watson and John Conway in classification of lattices connected to Niels Henrik Abel's arithmetic investigations. Modular forms and representation numbers connect to results by Srinivasa Ramanujan and Goro Shimura; these connections are studied at institutions including Harvard University and University of Cambridge.

Invariants and Arithmetic Theory

Key invariants include determinant, discriminant, signature, and the Hasse invariant; their study builds on foundations set by Carl Friedrich Gauss in his Disquisitiones and extended by Helmut Hasse and André Weil. Genus and class groups of quadratic forms interact with ideal class groups of number fields as in the work of Richard Dedekind and Ernst Eduard Kummer, while representation numbers and theta series link to Bernhard Riemann and Harvey Cohn-style analyses. Automorphism groups of forms relate to arithmetic groups studied by Marcel Berger and Armand Borel; mass formulas and Siegel's work reflect contributions by Carl Ludwig Siegel and Harold Davenport. Local invariants use techniques from Jean-Pierre Serre's local class field theory and the reciprocity laws of Emil Artin.

Applications in Number Theory and Geometry

Quadratic forms are instrumental in solving Diophantine equations considered by Fermat and later by Pierre de Fermat's successors like Yutaka Taniyama in context with elliptic curves studied by Andrew Wiles. Lattice theory and sphere packing problems connect to John Leech's discoveries and the Kepler conjecture resolved by Thomas Hales. In differential geometry the metric tensor is a quadratic form central to Bernhard Riemann's geometry and to Albert Einstein's formulation of general relativity; analytic torsion and index theorems invoke contributors like Michael Atiyah and Isadore Singer. Coding theory and cryptography exploit quadratic residue properties developed by Adrien-Marie Legendre and Évariste Galois, with modern deployments in work at Bell Labs and research by Ronald Rivest.

Quadratic Forms and Linear Algebraic Groups

The study of automorphism groups preserving quadratic forms leads to orthogonal groups and spin groups, areas advanced by Élie Cartan, Hermann Weyl, and Claude Chevalley. Connections to representation theory involve Niels Henrik Abel-era questions and modern results by Roger Howe and Harish-Chandra. The classification of algebraic groups over fields uses Tits' theory influenced by Jacques Tits and links to buildings studied by Kenneth Brown. Exceptional groups, including work on triality and forms by Élie Cartan and Robert Langlands, interact with quadratic form invariants in Langlands program contexts promoted by Robert Langlands and pursued in seminars at Princeton University and Institute for Advanced Study.

Category:Quadratic forms