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| Arthur–Selberg trace formula | |
|---|---|
| Name | Arthur–Selberg trace formula |
| Introduced | mid-20th century |
| Field | Number theory; Representation theory; Harmonic analysis |
| Notable contributors | James Arthur; Atle Selberg; Robert Langlands; Harish-Chandra; Roger Howe |
Arthur–Selberg trace formula is an analytic identity that equates spectral data of automorphic representations with geometric orbital integrals for reductive groups. It originated in the work of Atle Selberg and was vastly generalized by James Arthur, linking subjects such as Langlands program, automorphic representation, harmonic analysis, representation theory, and number theory. The formula is central to modern advances connecting L-functions, trace identities, and classification of automorphic spectra across groups like GL(2), SL(2), GL(n), and classical groups.
Selberg introduced an early trace formula for the modular group and hyperbolic surfaces, inspired by analogies with the Poisson summation formula and the Weyl law. Work by Harish-Chandra extended invariant harmonic analysis on real groups, while researchers such as Roger Godement, Ilya Piatetski-Shapiro, Robert Langlands, James G. Arthur, and Gérard Laumon developed techniques for adelic and noncompact settings. Arthur produced a general noninvariant trace formula and later a stabilized version that tied into endoscopic transfer and the stable trace formula, drawing on contributions from Jean-Loup Waldspurger, Jian-Shu Li, Laurent Clozel, and Colette Moeglin.
In rough form the identity equates a sum over characters of automorphic representations (the spectral side) to a sum over conjugacy classes or double cosets (the geometric side). Arthur formulated a distributional equality on the space of compactly supported test functions on the adelic points of a reductive algebraic group G over a global field, using input from adelic methods, adelic points, and truncation operators inspired by Selberg trace formula. The full statement requires data such as Levi subgroups, parabolic subgroups, intertwining operators studied by Harish-Chandra and normalization factors related to intertwining operators of Langlands–Shahidi method.
The spectral side organizes contributions from discrete automorphic spectrum, continuous spectrum from Eisenstein series, and residual representations. It involves sums and integrals of matrix coefficients of irreducible unitary representations parameterized by automorphic representations of groups like GL(n), GSp(2), or SO(n). Key components include Plancherel measures developed by Harish-Chandra, spectral decomposition akin to Peter–Weyl theorem, and the role of Eisenstein series studied by Langlands and Ilya Piatetski-Shapiro. Arthur's work introduces weighted characters and truncation operators to control divergence, with normalization via intertwining operators and spectral transfer related to functoriality in the Langlands program.
The geometric side decomposes into orbital integrals over conjugacy classes, including contributions from elliptic, hyperbolic, and unipotent conjugacy classes. Orbital integrals are regularized using truncation similar to techniques by Atle Selberg and integration over adelic quotients like those arising in Shimura varieties and modular curve contexts. Important tools include local harmonic analysis at places studied by Roger Howe and Joseph Bernstein, weighted orbital integrals, and comparison of trace formulas via matching of test functions connected to transfer factors introduced in endoscopic theory by Robert Langlands and Kottwitz.
Stabilization of the trace formula led Arthur to formulate and prove results on the stable trace formula and endoscopic classification of automorphic representations. The process uses endoscopic groups, transfer factors, and comparison of stable distributions, integrating work by Robert Langlands, Richard Kottwitz, Jean-Loup Waldspurger, and Ngo Bao Chau. Stabilization underpins proofs of cases of functoriality and relates to the proof of the fundamental lemma, a milestone achieved by Ngo Bao Chau with foundations by Gérard Laumon and Ngô Bảo Châu's collaborators and predecessors.
The formula has been used to establish instances of functorial lifts, compare automorphic spectra across groups such as GL(n) and classical groups, and study automorphic L-functions including symmetric power and exterior square L-functions. It has implications for classification problems solved by Arthur for classical groups, links to the Ramanujan–Petersson conjecture in special cases, and contributes to progress on the Sato–Tate conjecture for families. The trace formula informs results in arithmetic geometry related to Shimura varieties, Galois representations via conjectures of Langlands–Rapoport, and reciprocity laws envisioned by Gerhard Frey and Andrew Wiles in contexts of modularity.
Concrete instantiations include Selberg's original trace formula for the modular group and Maass forms on PSL(2,R), explicit trace identities for GL(2), and computational applications to counting automorphic forms on congruence subgroups relevant to Hecke operators and Atkin–Lehner theory. Work by Harris–Taylor and computational investigations by groups studying explicit trace formulae for low-rank groups provide numerical verification of trace identities and comparison with predicted automorphic spectra for groups like U(n), SO(n), and GSp(4).
Category:Trace formulas