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| Period integral | |
|---|---|
| Name | Period integral |
| Field | Mathematical analysis; Algebraic geometry; Number theory |
| Notable examples | Betti cohomology, de Rham cohomology, elliptic integrals |
Period integral A period integral is a complex number obtained by integrating algebraic differential forms over cycles in topological spaces arising from algebraic varieties, linking Bernhard Riemann, André Weil, Alexander Grothendieck, Pierre Deligne, and Maxim Kontsevich. Period integrals connect classical objects such as elliptic functions, hypergeometric functions, Abelian integrals, gamma function, and zeta function while playing central roles in modern theories developed at institutions like the Institute for Advanced Study, École Normale Supérieure, and University of Paris. They provide bridges between Betti cohomology, de Rham cohomology, Hodge theory, motivic Galois groups, and conjectures proposed by Kontsevich–Zagier and Grothendieck.
A period integral is classically defined by integrating a rational differential form with algebraic coefficients over a homology class coming from an algebraic variety defined over a number field; prototypical examples include elliptic integrals studied by Niels Henrik Abel, Carl Gustav Jacobi, Srinivasa Ramanujan, and later analyzed by Karl Weierstrass, Ferdinand Georg Frobenius, and Adrien-Marie Legendre. Concrete instances are the complete elliptic integrals appearing in the work of Gauss, Joseph-Louis Lagrange, and Adrien-Marie Legendre, the beta and gamma integrals of Leonhard Euler, and classical integrals expressing values of the Riemann zeta function studied by Bernhard Riemann, Leonhard Euler, and Gustav Lejeune Dirichlet. Simple algebraic curves like those in Elliptic curve theory yield periods that are integrals of holomorphic differentials over cycles related to Mordell–Weil theorem contexts examined by Andre Weil and John Tate.
In algebraic geometry one integrates algebraic differential forms on smooth projective varieties defined over fields studied in the works of Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne; typical ingredients evoke the cohomology theories developed at Cartan Seminar, Institute des Hautes Études Scientifiques, and Princeton University. Periods arise from pairing singular homology classes described by Henri Poincaré and Luitzen Egbertus Jan Brouwer with algebraic de Rham classes introduced in the frameworks of Jean de Rham and refined in Grothendieck's theory of crystalline cohomology, Fontaine's p-adic studies, and later expositions by Kazuya Kato, Luc Illusie, and Alexander Beilinson.
Cohomologically, periods are matrix entries of the comparison isomorphism between Betti cohomology and de Rham cohomology for varieties over Q or number fields, a perspective formalized by Pierre Deligne and Alexander Grothendieck. Hodge-theoretic structure studied by Wilhelm Hodge, Phillip Griffiths, James Carlson, and Claire Voisin gives period domains and variations of Hodge structure that appear in the investigations of Mumford–Tate groups, Griffiths transversality, and moduli problems treated at Harvard University and Princeton University. Period matrices for complex tori and Calabi–Yau varieties figure prominently in work by Philip Candelas, Burt Ovrut, and Edward Witten on mirror symmetry.
Periods of algebraic varieties are conjecturally controlled by motivic structures proposed by Alexander Grothendieck and formalized in Grothendieck's period conjecture and the theory of motives advanced by Yves André, Uwe Jannsen, Vladimir Voevodsky, and Pierre Deligne. Examples include periods of K3 surfaces, Calabi–Yau manifolds, and Abelian varietys with complex multiplication studied by Goro Shimura, Yutaka Taniyama, and Gerd Faltings. Relations among periods are predicted by the action of hypothetical motivic Galois groups in the spirit of research at Institut des Hautes Études Scientifiques, Max Planck Institute for Mathematics, and collaborative programs led by Pierre Deligne and André Weil.
Explicit computation of periods uses methods from classical analysis, algebraic geometry, and computational algebra implemented in software environments influenced by work at Massachusetts Institute of Technology, University of Cambridge, and ETH Zurich. Techniques include reduction to hypergeometric forms studied by Gauss and Norbert A’Campo, evaluation via Picard–Fuchs differential equations developed in the tradition of Picard and Fuchs, lattice reduction relating to Mordell–Weil lattices examined by John Tate, and algorithms inspired by David Hilbert’s work on invariant theory. Explicit formulas appear in the analysis of modular forms from Srinivasa Ramanujan and Goro Shimura, in closed forms for beta and gamma integrals due to Leonhard Euler and Adrien-Marie Legendre, and in period relations computed by Don Zagier and Maxim Kontsevich.
In number theory periods connect special values of L-functions studied by Bernhard Riemann, Atle Selberg, and Andrew Wiles to arithmetic invariants like regulators in Beilinson conjectures, linking to results by Kazuya Kato and Geordie Williamson. In mathematical physics periods control amplitudes in perturbative quantum field theory via Feynman integrals investigated by Richard Feynman, Kurt Symanzik, and Stuart Parke, and they appear in mirror symmetry and string compactification analyses by Edward Witten, Cumrun Vafa, and Philip Candelas. Periods also inform the study of modularity in contexts related to the Taniyama–Shimura–Weil conjecture proved in part by Andrew Wiles and Richard Taylor.
The study of period integrals traces from Leonhard Euler and Carl Friedrich Gauss through the 19th-century analyses of Bernhard Riemann and Niels Henrik Abel to 20th-century foundations laid by Alexander Grothendieck, Pierre Deligne, and André Weil. Landmark results include the formulation of period mappings by Wilhelm Hodge, the period conjectures of Grothendieck and the Kontsevich–Zagier period conjecture articulated by Maxim Kontsevich and Don Zagier, deep advances in transcendence theory by Alan Baker and Gerd Faltings, and explicit arithmetic applications culminating in the proofs of modularity theorems by Andrew Wiles and Richard Taylor that link periods to elliptic curve L-values.